Are there credit unions, or other non-bank financial institutions available in Mexico, to the general public?
Yes, of course there are. In Mexico they're called caja popular and one of the largest is Caja Popular Mexicana. There are also microfinance institutions (like Compartamos) but Mexico is famous for charging the highest interest rates in the world (well over 100%).
For more information on lenders (and ratings plus a really cool APR calculation tool) visit Microfinance Transparency.