If this behavior is affecting his performance in school, then call the administrator in the school and discuss it with him or her. Tell the principal or VP that you want to refer your son for Child Study because you suspect an emotional disturbance that might be affecting his school performance and his peer relations. They will get the school counselor and psychologist involved and do a school-based evaluation. Then you will be invited to the meeting at the school so you can participate in an intervention meeting. Your son may be eligible for services through 504 (if he is depressed), or if he is diagnosed with an Emotional Disturbance, he can qualify for special education services.
No, it's not normal. You don't have to have insurance to take him to the pediatrician, you would just have to pay out of pocket. I don't think this would be a terribly expensive visit involving lots of tests.
It sounds like he is either very depressed, or somehow he gets a lot of attention in life for being a crier, and I don't think this is expensive to figure out.
Best wishes.