Usually, you can follow links or do a google search to find the original text of the Call for Proposals. The full text of those almost always provides clear criteria for who is eligible. Almost always (probably unless otherwise stated), the PI has to be from a US institution, but contrary to the other answer here, I am not familiar with the PI needing to have any particular academic status (especially not tenured already as an associate prof, and certainly all the kinds of non-TT research scientist positions), provided they have a position which their institution officially lists as being eligible to be sponsored as a PI. Sometimes, a post-doc, lecturer, or adjunct prof might be eligible. Remember, the grants go to the institution primarily, and the PI merely executes them on their behalf. The granting agencies defer the responsibility for administering who is legally OK to conduct the research to the accredited institution. Policies can vary between institution, so you'll need to ask your local administrators.
Certain kinds of award from private agencies or those who work with sensitive information, e.g. DARPA, DOD, or DOE, might require you to be a US Citizen (not even just a LPR), but that should be clearly listed. I am pretty sure that "unrestricted," in this context, means that there is no such restriction on immigration status.
You are always free to shoot an email or call to the cognizant program officer for a particular grant, to ask for clarification. They will expect you to have read the full, original CfP, unless you want to put them in a bad mood that might bias the review of your prospective application!