Far be it from me to be answering GM setting stuff but the concept of a wandering apothecary is definitely not beyond Elder Scrolls. Healers, whether they be alchemists, restoration mages, clerics of the Temple, or conventional doctors are prized professionals and would be welcomed, even sought after by not just adventuring parties but professional organizations as well. In Morrowind (which is admittedly distanced from this thread's setting) the Tribunal Temple even mandated that its members go out and heal the sick and help the poor as doctrine, so there's lots of ways to do that type of character in universe. Hope that helps, and of course, the GM is the final say, I don't mean to tread on toes.