My seven year old has a hoarse cough and his chest hurts when he has these coughing fits? Should he see his doctor or wait it out? Thank you

Depends on timing. If he has only had symptoms for a day or two then there is a high probability he has a viral upper respiratory tract infection. In that case his symptoms should improve within a few days. If he is having trouble breathing or if he has a high fever or if he is unable to eat or drink, these are all reasons to take him to the doctor right away.
Be seen. It is likely bronchitis, but if he is having pain with breathing, it should always prompt you to have him seen.