Short answer: Lyme disease is not endemic to Australia. This is the conclusion supported by all credible recent scientific articles. This is also the Australian Government's official position.
Note: All bolded text is added emphasis not present in the original source. All cited scientific articles are peer-reviewed.
Given that the question was asked over 4 years ago, new developments have surfaced.
In 2019, Dehhaghi et al. wrote (in an open-access peer-reviewed article):
Importantly, there is no convincing evidence for the presence of locally acquired Lyme disease in Australia.
The presence of Lyme disease (Lyme borreliosis, LB) or Lyme-like disease in Australia is highly controversial. The Australian Government Chief Medical Officer convened a Clinical Advisory Committee on Lyme Disease in 2013 to advise on aspects of Lyme disease in Australia.
Hence, there is no evidence for transmission of B. burgdorferi sensu lato complex with Australian ticks. Whilst patients in Australia with Lyme-like disease may occasionally have positive Lyme serology, finding the causative agent using PCR or direct culture is regarded as mandatory for confirmation of local acquisition of infection.
The Australian Government report is available here and is extremely indecisive.
At the same time, the department is aware that Australian ticks are important vectors of human disease, e.g., rickettsial infections, Q fever and the newly described mammalian meat allergy.
The department will continue to encourage, identify and make known relevant Australian research on tick-borne disease.
The Australian Government petition statement states
Classical Lyme disease is an infectious disease that can be transmitted to humans if bitten by a tick carrying Borrelia burgdorferi sensu lato. In Australia, this organism has yet to be identified in Australian ticks or any another vector that could transmit disease to humans. It is for this reason that the Australian Government does not support the diagnosis of locally acquired Lyme disease in Australia.
Thus, the Australian Government is saying that Lyme disease is not endemic to Australia. This concurs with the New South Wales fact sheet. The Australian Parliament conducted a short-term inquiry into Lyme disease in Australia. Submissions (some are accessible) are available here. The final report, which "will be presented here as soon as it is available." is still not available, despite a reporting date of 20 June 2016. The interim report released on 4 May 2016 literally contains no conclusions and "recommends that the Community Affairs References Committee continue its inquiry into this matter in the 45th Parliament." Do note that the inquiry has most likely lapsed: "Inquiries that were not completed have lapsed and submissions cannot be received."
In 2016, Chalada et al. wrote (in an open-access peer-reviewed article):
In the last twenty-five years there have been over 500 reports of an Australian Lyme-like syndrome in the scientific literature. However, the diagnoses of Lyme Borreliosis made in these cases have been primarily by clinical presentation and laboratory results of tentative reliability and the true cause of these illnesses remains unknown. A number of animals have been introduced to Australia that may act as B. burgdorferi s. l. reservoirs in Lyme-endemic countries, and there are some Australian Ixodes spp. and Haemaphysalis spp. ticks whose geographical distribution matches that of the Australian Lyme-like cases. Four published studies have searched for Borreliain Australian ticks, with contradicting results. The cause of the potential Lyme-like disease in Australia remains to be defined.
[Summaries of the four studies]
2.6.1. Wills and Barry 1991
At least four of the spirochaetes isolated shared antigenic epitopes with B. burgdorferias demonstrated by ELISA, immunofluorescence and western blotting, suggestive of Borrelia species. However, details of the laboratory methods are not published and the organisms recovered were not made available for confirmation by another laboratory, rendering the experiment unable to be replicated.
2.6.2. Russell et al. 1994
The study found “no definitive evidence for the existence in Australia of B. burgdorferi the causative agent of true Lyme Borreliosis, or for anyother tick-borne spirochaete that may be responsible for a local syndrome being reported as Lyme disease”. The authors observed Wills' and Barry's cultured Borrelia and found them to be identical to their own SLOs, concluding that Wills' and Barry's cultured SLOs were also contaminant flagella aggregates. Russell et al. also had the advantages of Borrelia genus-specific PCR and a much larger sample size over Wills' and Barry's study. The conclusion of Russell et al's study–that no spirochaetes were able to be identified through culture or molecular methods in Australian ticks–therefore seems more plausible than the conclusions of Wills and Barry.
2.6.3. Gofton et al. 2015a
This work provides further evidence that the cause of the Lyme-like illness in Australia may not be a member of the B. burgdorferi s. l. complex. The finding of a novel relapsing fever Borreliain an Australian monotreme does provide evidence for the presence of Borreliae in Australia, but it is not known if this organism can infect humans, and should it do so, it is likely that it would present as a relapsing fever illness rather than with Lyme-like symptoms. These factors limit the likelihood that this novel Borrelia species is the cause of the Lyme-like illnesses seen in Australia.
2.6.4. Gofton et al. 2015b
A Borrelia genus specific flaB nested PCR was also performed on all ticks recovered. None of the ticks concerned yielded any Borrelia sequences or PCR products.
What this means (from the above excerpts and my own reading of the provided sources):
The first study (pay-walled) found epitopes of the Lyme disease-causing species in (4 out of 177) Australian ticks. Thus, this is evidence of Lyme disease endemic in Australia. However, Chalada et al. note no follow-up was conducted and the experiment is "unable to be replicated" due to lack of details of experimental methods.
The second study, with a larger sample size of "approximately 12,000 ticks" found "no definitive evidence" of the Lyme disease-causing species. The authors used genus-specific PCR and Chalada et al. seem to place more weight on this study.
The third study found a new species in the genus of the Lyme disease-causing species in Australian ticks. Notably, Grofton writes (in a different article) "The zoonotic potential and pathogenic consequences of this novel Borrelia sp. are unknown at the current time." Chalada et al. support that this novel species doesn't cause Lyme disease (though this isn't explicitly stated in the original study).
The fourth study (of 460 ticks) found no evidence of the Lyme disease-causing species despite a genus specific PCR.
Non-tick studies have also been conducted. In 2017, Irwin et al. studied 555 dogs and found no evidence of the Lyme disease-causing species.
To date, there has been no convincing evidence for locally-acquired Lyme borreliosis on the Australian continent and there is currently a national debate concerning the nature and distributions of zoonotic tick-transmitted infectious disease in Australia. In studies conducted in Europe and the United States, dogs have been used as sentinels for tick-associated illness in people since they readily contact ticks that may harbour zoonotic pathogens. Applying this principle, we used a combination of serological assays to test dogs living in tick ‘hot spots’ and exposed to the Australian paralysis tick, Ixodes holocyclus, for evidence of exposure to B. burgdorferi (s.l.) antigens and other vector-borne pathogens.
Except for a single dog presumed to have been exposed to Anaplasma platys, infection with Anaplasma spp. B. burgdorferi (s.l.), Ehrlichia spp., and Dirofilaria immitis, was not detected in the cohort of Australian dogs evaluated in this study. These results provide further evidence that Lyme borreliosis does not exist in Australia but that cross-reacting antibodies (false positive results) are common and may be caused by the transmission of other tick-associated organisms.
Collignon et al. wrote a narrative review (review article) about the subject:
There is no convincing evidence that classic Lyme disease occurs in Australia, nor is there evidence that the causative agent, Borrelia burgdorferi, is found in Australian animals or ticks.
It is a nice, short read, containing a section titled "Background to the belief that Lyme disease exists in Australia." and references to the Senate inquiry.
In 2018, Brown wrote:
Patients diagnosed with Lyme disease in Australia display a symptomology similar to ‘medically unexplained physical symptoms’ syndromes, experience social and financial harms, and are at risk of nosocomial harms. Negative medical interactions and the media may contribute to patients seeking alternative and potentially non‐evidence‐based diagnoses and treatments.
Beaman also touches on a similar concept in a 2016 article:
Despite this, patients who have not left Australia pay many thousands of dollars for non-specialist consultations and testing at overseas laboratories. Unproven long-term therapy with multiple antibiotics has resulted in serious complications, including allergies, line sepsis, pancreatitis and pseudomembranous colitis. Studies have shown that LD vectors are not found in Australia, and Lyme Borrelia has not been found in Australian vectors, animals or patients with autochthonous illnesses.
Is there an existing risk of contracting locally-acquired Lyme disease within Australia?
From recent studies: no, Lyme disease is not endemic within Australia. However, there is a conflicting 1991 study that found evidence of the Lyme disease-causing species (within 4 out of 177) native Australian ticks.
Note: The only recent peer-reviewed article I found that mentions Lyme disease exists in Australia is this one. I would like to point out this journal is less credible than the ones I cited because it is the kind that is open access and charges the authors a fee ("550 CHF (Swiss Francs)") to publish. Additionally, I can't find many references to this journal online.