If you were to artificially inseminate a human egg with human sperm, then implant that egg in the womb of, say, a chimpanzee, would that egg be able to grow into a human fetus regularly?

It's certainly highly unethical...

Unlikely. It's apparently been tried, and failed. There are a few factors that make it unlikely, one being that a chimps incubation period is 7-8 months, where a human is closer to 10 months. Another is the chimps body rejecting a human fetus. Although chimps are very similar to us, there are still many differences, all of which would make it unlikely that this could happen. But unlikely doesn't mean impossible.

Having said that, ethically it would be a nightmare.