An increasing number of funding agencies require recipients to make all research papers it funded open access.
Does any scholarship require the benefiting student to make open access any research paper they wrote during the scholarship period?
[...] We have adopted an Open Access policy that enables the unrestricted access and reuse of all peer-reviewed published research funded, in whole or in part, by the foundation, including any underlying data sets. [...]
These are the two examples I remembered directly.
Yes. Most research funded by the US Government (both internally by government employees and externally by outsider researchers receiving funding) is supposed to be available for the public. This would cover both direct scholarships/fellowships as well as grants and contracts to Professors that cover graduate assistantships.
This is not open access per say, because the research can be published in closed-access journals, but then is made public after an embargo period on a US Government page or publisher's webpage. The Association of Library Reseachers discuss this and has more details about specific agencies. Additionally, Nature News has a press release on the topic. Last, Columbia's Library also has an informative webpage on how different Federal Agencies meet the open access requirement. From an academic practitioner's perspective, this is probably the most informative page of the three I listed.
As final point, to revisit the "supposed to" caveat. Not all US funded research is covered (e.g., sensitive/classified Defense research). Some agencies may have not setup their methods for storing and releasing publications. Different agencies may enforce the open access policy with different levels of strictness. As a final caveat, most of the open access requirement policy is driven by an Executive Order rather than a Congressional mandated law. This makes the policies more easily subject to change.