My 4 month old has been sick for 3 weeks...?

Call your doctor and tell them it progressed to vomiting. I also had this problem with my son, but to find out he was teething which can cause cold like symptoms. They also develop more saliva which can cause them to gag and vomit. He popped his first tooth out 1 day shy of 5 mos. But that was my case, I'd call the doctor and if he doesn't do anything, I would say it is time to find a new pediatrician.