tl;dr- This answer is primarily an attempt to establish the context of the quote to determine its meaning. No conclusive sources were found, however:
It seems dubious that Glen Dettman actually made that statement.
If Dettman did make that statement, it seems likely to have been in the context of the Australian population pre-1938, not the global population pre-1980.
In that probable context, this claim seems fairly plausible.
The quote is misleading-at-best in the apparent context one might infer from just looking at the picture.
UPDATE: Found a source (printed-pages 361-362) that claims that most Australians weren't vaccinated against smallpox in the early 1900's. Coupled with the earlier point that the last smallpox case was in 1938, it seems that the alleged claim is probably about right in the context of the Australian population.
1. Tracing back the source of the quote
As far back as I've been able to trace the quote, it comes from an anti-vivesection (animal dissection for medical research) testimonial compilation (1989):
From the article "The Basic Anatomical Element: Bechamp's Microzyma" by Dr. Glen Dettman, AMM, BA, PhD and Archie Kalokerinos, MD, in Health Consciousness, Oviedo, Fl., Oct 1986: "It is pathetic and ludicrous to say we vanquished smallpox with vaccines when only 10% of the population were ever vaccinated."
-"1000 DOCTORS (AND MANY MORE) AGAINST VIVISECTION" (1989) [alternative link: PDF]
The trail seems to end here for now. Google isn't yielding any results for the alleged article,
The Basic Anatomical Element: Bechamp's Microzyma
, except in the context of the book claiming that the article exists.
Additionally, there seem to be inconsistencies with that alleged article when comparing it to Wikipedia's list of Dettman's publications:
The alleged article is from 1986, though Dettman's publication list stops in 1983.
The title of that article doesn't seem consistent with Dettman's prior publication titles.
The journal doesn't seem consistent with Dettman's prior publication venues.
This source lists Dettman's titles with both "BA" and "PhD"; it's incredibly strange to list a Bachelor's degree along with a PhD.
- This may be done in some places.
None of these concerns are conclusive, though the quote's attribution does seem dubious.
2. If real, the context may've been the Australian population
Glen Dettman lived in the pre-globalization days in Australia. His childhood, education, and professional career were all in Australia, and his publications seem to largely be in journals that explicitly have "Australia" in their titles.
And as the earliest-found source of the quote claims, Dettman's work was largely with Australians and the problems specific to Australia:
In Australia, Dr. Archie Kalokerinos, M.D., and Dr. Glen Dettman, Ph.D., discovered that some 500 out of every 1,000 Aboriginal children were dying in the Northern territories. The cause was a type of toxic shock reaction, complicated by vitamin C deficiency, brought on by immunization. In a two year period without vaccination and with improved nutrition not one child died.
-"1000 DOCTORS (AND MANY MORE) AGAINST VIVISECTION" (1989) [alternative link: PDF]
Given this background, along with the plausibility of the quote in this context (discussed in the next section), it seems likely that, if real, the quote may've been in the context of Australia.
3. In the probable context, this claim seems plausible
According to an Australian government health website, smallpox hasn't appeared in Australia since 1938:
Smallpox last appeared in Australia in 1938, and the last case naturally occurring cases in the world was reported from Somalia in 1977. The virus is now only held officially in two secure laboratories. Any reappearance of smallpox is likely to be the result of bioterrorism, but the risk of this is extremely low.
–"Smallpox (Variola)", NSW Government Health [PDF]
And though Australia hadn't seen a case since 1938, the World Health Organization (WHO) claims that smallpox was still raging worldwide in 1950, not being seriously pushed back by vaccination efforts until about 1967 before being declared eliminated in 1980:
In the early 1950s – 150 years after the introduction of vaccination – an estimated 50 million cases of smallpox occurred in the world each year, a figure which fell to around 10–15 million by 1967 because of vaccination.
In 1967, when WHO launched an intensified plan to eradicate smallpox, the "ancient scourge" threatened 60% of the world's population, killed every fourth victim, scarred or blinded most survivors, and eluded any form of treatment.
–"Smallpox", World Health Organization (WHO), via archive.org
Given this timeline, it seems plausible that serious vaccination efforts didn't occur in Australia before smallpox was eliminated there by 1938. Therefore, it seems plausible that the explanation for smallpox being eliminated in the Australian population may not involve most of the population getting vaccinated.
Though to stress it, this is merely plausible given the limited information found so far. This answer does not claim that there was no widespread (>10%) vaccination effort in Australia pre-1938, only that evidence for such an effort hasn't popped up yet.
UPDATE: Found a source that discusses the prevalence of smallpox vaccination in Australia around that time. As speculated, it appears that vaccination wasn't applied to most of the population.
The full quote since it contains a lot of relevant information (shortened version below):
SMALLPOX IN OCEANIA DURING THE 20TH CENTURY
Apart from some widespread outbreaks among the aborigines of Australia during the 19th century, all of which died out after a few years (see Chapter 5), smallpox never became established as an endemic disease in Australia, New Zealand or the islands of the Pacific Ocean. From the beginning of the 20th century they were well protected from importations by their geographical remoteness and the effective quarantine measures imposed on visiting shipping. This situation led to a disregard for infantile vaccination, which by the early years of the 20th century had reached a very low level in Australia and New Zealand. Nevertheless, a long-standing requirement for valid vaccination certificates for all travellers, combined with vigilant seaport and, later, airport medical inspections and quarantine, kept both Australia and New Zealand free of serious outbreaks of smallpox, except for separate importations of alastrim into each country in 1913.
The Australian epidemic of alastrim was initiated in April 1913 by a ship’s steward who was infected in Vancouver, Canada, and slipped through the medical inspection in Sydney (Cumpston & McCallum, 1925). The outbreak which followed lasted until December 1917 and produced 2400 cases in various parts of Sydney and in country towns in New South Wales, but only a minor extension into one other state-Queensland. It was of very low virulence, with only 2 deaths attributable to smallpox, and of low infectivity, spreading slowly in a largely unvaccinated population of 1.8 million. Control was achieved by vaccination and the segregation of cases and contacts. Subsequently a few very small outbreaks of both variola major and variola minor occurred, but the effectiveness of the quaran- tine arrangements was indicated by the fact that, between 1909 and 1923, 40 ships were quarantined for smallpox or suspected smallpox.
–"Chapter 8: The Incidence and Control of Smallpox between 1900 and 1958" (printed-pages 361-362)
The relevant part appears to be:
The outbreak which followed lasted until December 1917 and produced 2400 cases in various parts of Sydney and in country towns in New South Wales, but only a minor extension into one other state-Queensland. It was of very low virulence, with only 2 deaths attributable to smallpox, and of low infectivity, spreading slowly in a largely unvaccinated population of 1.8 million.
–"Chapter 8: The Incidence and Control of Smallpox between 1900 and 1958" (printed-page 361)
This source makes it sound like vaccination was deployed strategically to high-risk demographics, e.g. travelers and those near outbreaks, though it seems to claim that most of the population wasn't vaccinated.
The evidence so far reflects:
The quote's attribution seems dubious.
If the quote is real, it may've been in the context of the Australian population and how they wiped out smallpox by 1938, not in the context of the global population in which major vaccination efforts were pushed after 1950.
The limited information found so far leaves the claim that smallpox was eliminated in Australia without more than 10% of the population getting vaccination plausible.
Finally, it should be understood that while the original claim, if real, may've been defensible, a misrepresentation of it taken outside of a context in which it may've been defensible doesn't necessarily enjoy the same validity. Of course the smallpox vaccination attempts were far more crucial to wiping out smallpox worldwide.
From what I've read of Dettman's work, it sounds like he was an actual scientist who did good medical research that helped folks out. If some of these sources are to be believed, it sounds like Dettman did find that vaccines were causing harm to some Australians, which seems like a plausible finding. It remains understood that even modern versions of the smallpox vaccine can have rare-but-serious side effects.
It should be noted that Dettman's work was in the context of the mid-1900's at which time medical understanding and technology wasn't what it is today. His findings and statements, where real, should be understood as historical; they don't necessarily reflect on modern medical realities nor on modern political concerns. Those should be put aside entirely.
All of that said, while Dettman's possible claim may've been defensible in a seemingly likely context, of course such a claim would be misrepresented if taken outside of that context. So, internet memes and modern political arguments based on such misrepresentations should be viewed skeptically.