Are there any funding sources available for OA/APC charges? I'm a self-funded part-time PhD student. My (UK) university does not provide funding for OA charges/APC for self-funded students. I had a few situations now where I've struggled to come up with funding for charges. Up until now my supervisor was able to help through grants he held.
This is getting a bit awkward and doesn't seem sustainable though. I've spoken to my department and library and they couldn't help. Is there any source of funds for APC that I could apply for in my situation?
The real problem here is your university.
Let me just point out (although you may already know this, other might not), your supervisor was almost certainly not paying OA charges with their grant, as almost all UK funders prohibit it. UK funders give a block grant directly to the university library in order for them to pay OA charges. Different universities have different rules about what they will and won't fund, but generally an application to be funded has to be attached to a grant code, even if the money isn't coming out of a grant.
As for suggestions - I don't really have any. If you've tried your library, and you've tried your department, you sort of out of luck. In this situation, there are two possibilities:

Apply to the journal to have fees wavied on the basis of being unfunded. Its a long shot that is usually reserved for researchers from places with underfunded university systems.

Submit to a journal that doesn't charge OA/APC fees. Although this is unlikely to be an OA journal, at most journals you have the option of either pre-printing, or putting the final reviewer authorized manuscript (but not the formatted proof) on an archive server, either a personal one, a university one, or a discipline-wide one (e.g. PubMed Central).