Can the mutation rate vary for individuals of the same species, growing in similar environments?
Yes.

between individuals: https://academic.oup.com/mbe/article/34/2/419/2528250
between individuals: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4617969/
hypermutation (transient): https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1087688/