Depending on what you mean by open access, all NIH grants already do come with such requirements. Any publications resulting from NIH-funded research must be deposited, in full text, including figures, into PubMed Central.
Open science/sharing the data can work as a voluntary movement, it has not been captured by private interests, but I'm really interested in seeing Open Access pan out before that. The current policy, if I'm reading it right, is saying that the manuscripts must be submitted to PubMed Central, but PMC or the researcher (whoever decides to trigger it first I imagine) only has an obligation to release them after twelve months. If PubMed Central doesn't release the research instantly and the researchers aren't aware of their rights or are subjet to some pressure/contradicting (possibly illegal) agreements, there's a loophole in the policy as written.