tl;dr; Yes, but that in no way implies the vaccines are ineffective at one year because there are immune responses besides simple antibody count (which is what prevents initial infection).
Yes, BUT (at least referencing the mRNA) vaccines have been shown to be highly effective in preventing severe illness as a result of infection:
The data, involving nearly 12,000 people who have been followed for at
least six months after vaccination, also showed the two-dose shot was
100% efficacious in protecting them from severe disease as defined by
the U.S. Centers for Disease Control (which means any disease
requiring hospitalization, intensive care or a ventilator) and 95%
efficacious in protecting against severe disease by the FDA’s broader
definition, which includes any respiratory distress or shortness of
breath at rest, or oxygen saturation below 93%.
While this study only provides 6 months of information, longer studies on the immunity gained from natural infection, combined with studies on immune response comparisons between natural infection and vaccination, allow for logical deduction/an extrapolation of data sets around the vaccines efficacy at the one-year post-inoculation time frame. Without empirical data over the exact time frame firm statements are hard to make, but given the evidence suggests long term protection well beyond a year at this point, in tandem with the reasons outlined below, I believe the one year time frame is currently safe to discuss in an evidence-backed manner.
Information from related viruses such as SARS-COV1 (immune memory lasting 17 years after initial infection) and immune responses to other pathogens such as the flu implies that even though the vaccine may not prevent an initial infection for a long term after initial inoculation, the vaccines should provide some lasting protection as they produce "memory b" and "memory t" cells, which have been shown to last years, decades, and even lifetimes. This study confirms natural immune responses are sufficient for up to eight months, and with extrapolation of the data sets, a true long term protection from the virus(well over 1 year, unless cell half life suddenly and drastically decreases in an unprecedented manner). Given the vaccines have been shown to induce b and t cells to at least the same degree as natural infection, it follows there will be a similar long term protection from severe illness.
While the use of naturally occurring protection from those previously infected is a good baseline to help establish long term vaccine efficacy though comparison of early vaccinated vs. recovered patient data and longer term recovered patient data, it should not detract from the apparent value of the previously infected being vaccinated. The patients protection is increased by vaccination, and more homogenous (in sufficient levels of protection) compared to the resulting protection only from natural infection. Although other studies indicate natural protection is sufficient to prevent severe reoccurrences in well over 90% of study participants, as linked above, there are still potential benefits versus new variants to being vaccinated regardless of prior infection. Vaccine efficacy versus new variant strains is still a hotly debated topic, and the information provided regarding the effectiveness against new covid variants is certainly up for debate more so than the efficacy of the vaccines against the initial covid variant. It's important to note the evidence presented here is strictly related to the initial variant as there isn't sufficient data around the new variants at this time - although all preliminary findings show the vaccines offer at least some protection against currently known variants of concern as defined by the WHO. However, the CDC hasn't published as certain of statements on the matter (see last section).
While the vaccine may not prevent another initial infection, it should, in most cases, provide a level of protection adequate to slow or potentially even stop the larger spread of the virus for at least one year after inoculation, with current data suggesting an even longer time frame.
Sources, 1, 2, 3, besides that linked in first word, confirming new initial infections after vaccination(also stating experts confirm it’s not a concern or a knock on vaccine efficacy). Also note the legitimacy of the information from the initial source that raised the question, both in terms of an inability to prevent new infections, but also summarizing most of the information here far more eloquently.
Thanks to @mmmmmm, and @LangLangC for challenging key points and statements made in haste within the answer. More critique / counter points to improve the answer are warmly welcomed.