Yes for bacteria, maybe for viruses, probably not for prions. For what it's worth, I'm LDS (Mormon) and I think my view is probably pretty common amongst members of the church. That said we would technically use the term spirit instead of soul, I just used soul because it's a more commonly understood term. Basically all living things have a spirit in addition to their physical form.
So when you transplant whatever thing from a body to another body, what happens with its soul/spirit? Notice that we transplanted all of them (in humans and animals). Do we have multi spirit/soul chimeras?