vaccination
You should go to your local Heath Department for vaccination. Tell them you need to get shots for the Immigration. They already have special form for the Immigration, for people who doesn’t have a shot record or unable to provide one. Also - it will cost you less if you do your shots at HD instead in the licensed doctor’s office.
Your husband won’t have to endure all those shots he missed as a child. He is a grown-up now and most of those shots doest apply for him anymore.
The nurse will ask him about kind of shots he had, but she will admit only what is specifically requested by INS and he doesn’t have a proof. Except for the TB. This one he has to do at the Immigration doctor’s office.
I never heard a restriction on doing your shots before the medical exam. People are getting shots every day.
I did my shots for the immigration couple of years ago. I didn’t see the doctor yet. They are supposed to send me a notice for the doctor’s exam later (I am adjusting my status based on the asylum approval). I didn’t have any records of my shots when I came to US (even though I had all of them as a child). Now, at least, I have that record and if during my health examination doctor will find I am missing one or two it would be only one or two more that I have to get. Same for your husband.