Does a critical mass of infected individuals exist after which mutations will overtake vaccination attempts?
Epidemiological modeling
If a virus is able to change so that it renders previous vaccination inefficient, reinfecting those who were previously vaccinated, one could describe this process using epidemiological SIS model (Susceptible-Infected-Susceptible) or its modification that includes a vaccinated group (like SISV model), and indeed estimate how quick should be vaccination in order to eradicate dthe virus before everyone gets infected. One could even add a D (deaths) compartement and study whether the virus may lead to extinction of the huiman species (unlikely, since the survival rate is very high).
Real viruses
In reality we are not dealing with such a situation, since there is no reason to think that mutations make SARS-CoV-2 able to overcome the immune defenses (although they do make it more contagious and more dangerous to non-vaccinated individuals).
To clarify the issue let me outline the scenarios where viruses do escape vaccine resistance.
Influenza
Flu is known to be susceptible to vaccination, and this is a lasting vaccination. What makes this virus to come back every year is gene reshuffling, where it exchanges some of its several genes with similar viruses living in animals (the roles of some of these animals in flu outbreaks gave rise to such names as swine flu, * birs flu*, etc.). This means that every few years we are dealing with an essentially new virus (rather than a virus that has accumulated many mutations), which is not visible to the antibodies created by previously administered vaccines. In this sense, one cannot exclude a possibility of a new coronavirus causing a pandemic in a decade or so, just like the current outbreak follows similar ones due to SARS in 2002 and MERS in 2012.
HIV
HIV is know to adapt very quickly to the new antibodies, evading the immune defenses. This makes it impossible (or at least very difficult) to develop a vaccine against this virus. However, HIV is much less contagious than flu or coronavirus, since it needs to infect specific cells/tissues. In orther words, it is stopped by the barriers of the immune system other than those boosted by vaccination. Coronavirus is a much simpler virus than HIV and will never reach the same degree of sophistication via mutations.
Bacterial pathogens
Bacteria are known for routinely developing resistance to antibiotics, which has been somewhat of a hot issue in terms of antibiotic development. They however share the same weakness as HIV - being less contagious (and far more complex).
To summarize, the devil is in the details: the scenario outlined in the question is good for science fiction, but it has very low probability to be realized in real life.