Not knowing the size/configuration of the built-in or the materials it is constructed with (vinyl, wood veneer, solid wood, etc.), I am a bit challenged to provide much constructive advice.
However, I did perform a search for cabinet/furniture refinishers and cabinet refacers in your general area.
- The refinishers I spoke with had very little, if any, knowledge of low-VOC finishes or FSC woods other than being California Air Resource Board compliant.
- This was coupled with a general lack of interest in your project (as described) because refinishing or replacing the doors on a small built-in-place is not terribly cost-effective and fraught with issues like the ability to match stain colors. (Depending on wood species, wood stains often do not penetrate evenly on previously finished pieces regardless of how much sanding you might do.)
There is one refinisher, who only refinishes (but does not replace) doors, that you could contact, and that is N-hance Wood Renewal in San Ramon.
Your second option is to reface the entire piece and replace the doors.
- This can ensure a better color match and more style options with respect to replacement doors.
- There is one refacing company in your area that is believed to use low-VOC finishes as well as FSC woods, but I was unable to confirm their methods or materials. They are Segale Bros. Wood Products.