We have 57 million fully vaccinated [0] people and 6 million infected/recovered [1] out of 83 million inhabitants. The recovered even have to get a shot later to keep their status as recovered/vaccinated (which exempts from the restrictions that are imposed now), so they might even be considered as a subset of the vaccinated group. More infections therefore would not have increased the total number of immune people but only cost more lives.
Beside that we can vaccinate a multiple of the number of newly infected people per day just fine. It's been about people being unwilling to get the shot, not availability of vaccines for quite a while now. Example: one day of vaccinations like yesterday (~1 Mio) [0] equals two weeks of new infections at levels like today (>73000).
[1] https://experience.arcgis.com/experience/478220a4c454480e823...