What time of year do STEM professors typically work on grant proposals?
Throughout the year. At least in the U.S., timings are driven in part by when funding agency deadlines are. For most NIH  grants, there are three submission windows per year. For NSF, there are one or two, but when these occur is different for different programs. Plus, in general, thinking about funding is a major task, and can't be confined to a brief span of time.