Where can I find a doctor for my child, if we have no health insurance?

Ask. Ask the nearest children"s hospital for resources. Or call the local pediatricians and see what can be worked out. Most states have programs in place to pay for vaccines and help with office visits. The pediatricians may be able to help.
Call office RHC/FQHC. Just about any pediatrician will make arrangements for a child who cannot pay. Ask for the office manager and be open and honest about your situation. Also, if a clinic is a rural health clinic, (rhc) or a federally qualified health center(fqhc), they must have arrangements for patients without resources/insurance. Or come to danville, il to my office - i'll be happy to see your child!
CHCs. Chcs, or community health centers, are private, not for profit organizations that receive government grants to take care of people who are uninsured or under-served. You can find a heath center near you at http://findahealthcenter.Hrsa.Gov/search_hcc.Aspx bring your income information when you come, and they will be able to adjust the costs accordingly. Good luck!