Postdocs can be the PI on individual postdoctoral fellowships (NIH F32 Individual Postdoctoral Fellowships, NSF Postdoctoral Fellowships, fellowships from private agencies, etc.) under the mentorship of a faculty member.
For all other grants, the Office of Sponsored Projects PI/PD Eligibility Policy states that, "At the recommendation of the appropriate school chair/department head, a postdoc employee may be designated as a co-PD/co-PI.”
Information Blurb:
Can a postdoc be a principal investigator (PI) on a grant?