It is not required that everyone be a U.S. citizen. You're correctly that your employees will need to be able to document they are able to legally work in the United States. However, there are numerous federal contractors and subcontractors that have employees in the U.S. who are foreign nationals here on H-1B or other visas.
You can use this OFCCP audit checklist to ensure you're doing what is required to maintain OFCCP's regulations including VEVRAA, Section 503, and EO 11246. Or request a demo to streamline your compliance and recruiting efforts.