Can students apply for Research Grants (US)?
Technically, in the US, most grants are not normally submitted directly by the principal investigator, but are instead sent in via a "grants office" that acts as the interface between the principal investigator and the funding agency.

Consequently, in addition to the grant-specific funding rules you have from a specific funding agency, you also have to deal with whatever restrictions your university grant-office has. For instance, it's not likely that your grants office will accept a proposal submitted by a graduate student instead of by a faculty member or similar personnel (lecturer, research professor, etc.).

In general, though, you need to read the requirements of the individual granting agency, as well as ask your local grants office.