Many museums are private, tax-exempt organizations and as such, they file an IRS form 990, which is publically available. They do not have to make their annual reports (which often contain information such as donor lists, attendance figures, etc.) public, nor must they make the sources of their funding public.
Finding Information on Other Funding Sources:
Special funding for specific exhibitions is credited on introductory wall texts and in exhibition catalogues, although those donors may or may not have provided funding for the entire exhibition.
Other sources of funding may be a bit more tricky to uncover. Members of a Museum's Board of Trustees make annual donations but the amounts vary widely from institution to institution. Members' publications often list donors under specific amounts, but you will not find these publications in a library (other than the museum's own library). Newspapers will cover major financial events (such as the financial rescue of MOCA a few years back). The library's LA Times database will be your best source for local museum news.
Many museums publish books that document the highlights of their collections. These are sometimes called collection catalogs and sometimes gallery guides.
The best way to find them is to use the Library Search and to search on the full name of the museum and add the word "catalogs"