TamiHi,
My SWCAH daughter was given all the veccines at the regular schedules, and she hasn’t got any problems. She received the diperte together with pertissus and tetanus. We gave her the varicella (chicken-pox) vaccine as well, (though in our country it is not part of the regula schedule) and she just took is wonderful - no side effects at all.
Our peditrician asked our endo whether my daughter can get all the vaccines, and she told yes, so I think you should not worry about it; ask your endo, she is most familiar with CAH, than a peditrician.
Kind regards, Tami