You're correct that it's possible for vaccines to be distributed in a way that promotes mutations[1]. You're also correct that there are many factors, including number of times the virus replicates and also selective pressure.

At this point, it looks like our mRNA vaccines:

- leave a short enough window between doses to minimize mutations

- reduce viral load within a vaccinated person[2]

- prevent many transmissions entirely[3][4][5]

...which, added all up, appears to net out the way you'd expect: the vaccine reduces the risk of mutations.

1. https://www.npr.org/sections/health-shots/2021/02/10/9659409...

2. https://www.nature.com/articles/s41591-021-01316-7

3. https://www.cdc.gov/coronavirus/2019-ncov/science/science-br...

4. https://www.physiciansweekly.com/covid-19-breakthrough-infec...

5. https://www.scientificamerican.com/article/the-crucial-vacci...