Does he have a history of sensory issues? The vest can be a really overwhelming experience to a kid that has a low sensory threshold (it's tight, it's loud, it SHAKES... i.e. some kids can't even tolerate a scratchy tag in their clothes; some feel all-over itching when the vest is on so imagine how that feels to a kid with that issue.) So I imagine some kids might translate discomfort or overload as 'pain' when trying to communicate this to parents, especially since pain is something we take seriously.
So I agree with the suggestions of experimenting with different clothes to wear underneath. Anything 'bumpy' (buttons, bulky collars, wrinkles, etc might be uncomfortable.) Emily always feels cold, oddly enough, when wearing hers. She doesn't like the air blowing out the sides- I don't know how she feels it since it blows away from her- but always wants a blanket on her. She screamed through her first session, hating the whole thing- and even after the screaming stopped she disliked it for a while and we had to do reward charts for a couple months to gain cooperation (she was your son's age too, btw.)
She has big sensory issues, so this is why this occurred to me as a consideration.
I'd also consider bringing the jacket to your clinic and make sure it's fitted properly to him; the RT or PT there should be able to check it over for you.
I hope you are able to get this figured out soon so this is a less traumatic experience for him asap!