Who should I call? Gutter installers won't know anything about
replacing that wood part, would they?

Yes, they should. We did similar jobs for years. These days though, I dont
know WHAT the gutter installers know how to do. Give 'em a call.
Depending on how "bad" the wooden fascia board is, it may be possible to
cover it with aluminum fascia covers. Either way, the gutters have to come
off.

I don't particularly need new gutters, so if I call a carpenter, do you
think they would take off and replace the gutter?

A local 'handyman' with a good reputation (ask around) might be the guy for
this job.