The government contracts with your company, not with you personally, so owning a registered company with valid work authorization allows you to pursue contracts. Permanent residents (green card holders) can register and win government contracts.
Citizenship matters mainly for special SBA set-aside certifications. Programs such as 8(a), HUBZone, and woman-owned certifications generally require that the company be at least 51% owned by a US citizen. A non-citizen owner can still pursue the broad market of contracts outside those set-aside programs.
If qualifying for a set-aside matters to you, ownership can sometimes be restructured — for example, a citizen family member or partner holding the qualifying majority of shares while decision-making control is defined separately in the operating agreement. The program advises on the right structure during your onboarding and strategy calls rather than leaving you to decide alone, and certifications that require citizenship can simply come later if and when citizenship is obtained.