Q : What forms are needed?

Form I-693.  If you are applying for adjustment of status in the United States, Form I-693, Medical Examination of Aliens Seeking Adjustment of Status, is used to report the results of the medical exam and vaccination assessment to USCIS.

Vaccination Supplement to Form I-693.   There is also a vaccination supplement which your civil surgeon will provide.  It is used to record the results of the vaccination assessment.