This could be many things (including a food allergy, irritable bowel, or many other things) but be sure at least to ask about "chronic appendicitis." Even some doctors don't realize that not all appendicitis is "acute appendicitis" that progresses rapidly and ends with the appendix being removed (or bursting). Appendicitis can be chronic, meaning it goes on and flares up over time, and can have these symptoms you describe.
I would insist that he go to the hospital for tests which probably will include an ultrasound and possibly an MRI. I would not "wait and see" this time but go to the ER. ERs usually expedite children when they come in so he should be seen almost immediately.
My daughter had appendicitis with almost no vomiting (one small vomit) and no fever until about two hours before the operation itself. She was given an ultrasound that did not show swelling typical of appendicitis and then given an MRI that also did not show typical swelling. But a surgeon said he was 99 percent sure that that's what she had -- and he was right. She had it removed the next day. She also never had the screaming, doubled-over pain that most of us associate with it -- until about one hour before the operation.
So appendicitis can take many forms. Don't let the doctor say that "because the pain's not focused on the right side, it's not that." My child's pain was right around her belly button, not on her right side. Again, this may be irritable bowel, twisted bowel, a food allergy even -- but you can't know without more testing than you can get from just one visit to the pediatrician. Hope your son improves soon and you get a diagnosis fast.