Since vaccination is not 100% protective against infection (i.e. the virus can sometimes succeed in establishing and replicating in a vaccinated host's body), the answer is yes. Furthermore, since vaccination is also not 100% protective against transmission — although it does lower the probability of transmission from infected individuals — such mutations could also be transmitted to other hosts.
However, vaccination greatly reduces the probability of infection and onward transmission. It might also be expected to reduce the duration and intensity (viral load) of infection, which would further reduce the probability that a mutation arises in an infected person (because the total number of mutations is proportional to the population size of the virus within the host).
Singanayagam et al. (2021) found that the peak viral load was not greatly reduced in infected, vaccinated people infected with the delta variant (relative to unvaccinated people), but that the decline from the peak was faster; with more effort/re-analysis of these data, we could estimate the magnitude of the change in the expected total number of viral replications over the course of an infection, which would in turn tell us the expected proportional difference in overall mutation probability due to changes in the within-host dynamics of the infection. (Also see this answer for more detail on viral load in vaccinated vs. unvaccinated individuals.)
Responding to a comment: the mutation rate (i.e. per base pair per viral replication, or per genome per viral replication), and the mutational spectrum (i.e. what kinds of mutations occur) are not expected to change much under vaccination - these are determined by the biochemistry of viral replication, and are probably not affected by what the host's immune system is doing.
(Disclaimer: my background is in ecology/evolution/epidemiology, not virology or immunology or biochemistry.)
Singanayagam, Anika, Seran Hakki, Jake Dunning, Kieran J. Madon, Michael A. Crone, Aleksandra Koycheva, Nieves Derqui-Fernandez, et al. “Community Transmission and Viral Load Kinetics of the SARS-CoV-2 Delta (B.1.617.2) Variant in Vaccinated and Unvaccinated Individuals in the UK: A Prospective, Longitudinal, Cohort Study.” The Lancet Infectious Diseases 0, no. 0 (October 29, 2021). https://doi.org/10.1016/S1473-3099(21)00648-4.