According to the Equal Employment Opportunity Commission (EEOC), yes. Simply requiring proof of vaccination by itself does not violate either the Americans with Disabilities Act (ADA) or the Genetic Information Nondiscrimination Act (GINA).
However, employers should be aware that the ADA limits their ability to make “disability-related inquiries,” and GINA generally prohibits employers from requesting employees’ genetic information. While neither of these laws prevents employers from asking for proof of vaccination, employers are wise to explicitly tell employees not to provide related medical or genetic information as part of the proof. For example, employees should be advised not to provide their full medical history or extraneous information about their medical visit when providing proof of vaccination. See above for a sample GINA blurb.