Polio is still endemic in Afghanistand and Pakistan and I believe there have been irregular detections elsewhere.
To OP, at this point it is not clear. If vaccination was available early in the pandemic and/or if the infection levels where kept at a low level until now, and if everyone was getting vaccinated, then the answer would have been a yes.
However, one should take a step back in understanding how eradication works. It is not necessarily just a matter of vaccination, but it is about creating a situation where an infected person is unable to infect enough folks to sustain pathogen spread.
Herd immunity could be achieved by a combination of vaccination, immunity, as well as social measures (e.g. distancing) for example.
But right now we have still over 14.8 million detected infections (and likely many more undetected) which is a huge reservoir for the virus and has a high risk of the creation of new variants. I have lost track of how many variants there are now in circulation, though only relatively few are classified to be of concern. Nonetheless, there is a big risk that for at least some of the variants, the vaccine will be less effective. We have already observed across the world that the UK variant (B.1.1.7) has displaced the wildtype. And this also affects how we should interpret vaccine efficacy data. Pfizer/BioNTech and Moderna efficacy data were generated earlier in the pandemic where mostly the wildtype was around. However, AstraZeneca already included data from patients with the B.1.351 (South African) strain, against which the vaccines might not work as well.
Some smaller data sets indicate for example that the Pfizer/BioNTech vaccine is about 89.5% effective against any infection with the B.1.1.7 variant and only 75% against B.1.351 in some groups (DOI: 10.1056/NEJMc2104974). However, they still protected with over 90% against severe diseases. The issue there is that while it prevents hospitalizations, it is still possible that folks get infected and may potentially transmit it to vulnerable persons.
Aside from variants we got the issue that in many (most) populations we will not achieve anything close 100% compliance. Surveys in various countries, including the US, UK and Canada, indicate that up to 35% of those surveyed indicated that they won't get the vaccine. Another big issue is worldwide-timing. If vaccines are only provided in richer countries, then those who cannot afford it are basically a reservoir for the virus. If we take another year to vaccinate them, it will be a full year where new variants can rise.
But even if just focus on local issues (and I want to emphasize that this would be really stupid to address a pandemic), we can do a little bit of a back-of-the-envelope calculation here. Let's focus on vaccination as the sole measure and let's assume we need ~80% immunity in the population to reach herd immunity.
Let us further assume that the vaccines have an effectiveness of 90%. In order to achieve 80% immunity, it would be necessary to vaccinate 89% of the population to reach the herd immunity target. Only few countries (according to polls) are at that level of theoretical compliance. If we use US polls as an estimate of vaccine willingness (~75%), we can estimate that with a 90% effective vaccine we will have ~68% immunity, lower than almost all current estimated requirement for herd immunity.
If the vaccine effectiveness goes down to 80% (due to variants for example) we would need to vaccinate every single person. So even if there are no barriers to providing vaccines to folks, it is tricky to rely on it alone to push down viral numbers to a degree that there is no net transmission.
So no, based on the current situation I actually do not think that the current rollout in practice is likely going to eradicate the virus completely. The more likely scenario (I believe) for now is that it will become endemic. However, the optimistic scenario is that it will be better controlled via regular vaccines (and treatments) and won't have the same horrible death toll in the future. It might indeed become something like a flu, which, I want to emphasize has regularly costed many lives and is not really harmless either (though compared to COVID-19 it is comparatively tame).
And as a minor sidenote, I would like to emphasize how behavioral changes have affected flu-related deaths. In the years prior 20-50k folks are estimated to have died each year from influenza in the US. Last year it was a few hundred reported so far.