Researchers here at Wayne State with H1B visa status are not precluded from submitting proposals to the NIH. Grants given are technically awarded to the university, so submission is allowed as long as the H1B holder is officially employed by Wayne State.
You will need to remain at WSU long enough to finish your proposed project, and you’ll have to state in your application that your visa will allow you to be here long enough to be productive. As you know, H1B visas are held for a maximum of six years, and it may be issued in increments of up to three years by the USCIS.
While submissions and award are both possible, there ARE special procedures for funded foreign nationals to perform select agents and toxins research. If your proposed project has an agent or toxin that is considered “select” (go here to find out: http://www.selectagents.gov/SelectAgentsandToxins.html) we’re happy to walk you through the whys/hows.