I assume this is not US-based? There are no a priori restrictions on foreign nationals being Principal Investigators on federal grants in the United States. You actually have other administrative roadblocks to consider as well. There isn't really such a thing as an "independent researcher" unless you are working out of your house, in which case, your odds of getting funding drop precipitously. Awards are made to institutions, not researchers. So the National Science Foundation (NSF) does not give money to your advisers per se. They are simply the technical contact for the project. They do not have the rights to the project that many imagine.
Every university will have a policy on who can receive funding. This is typically called "PI rights", and should be published online in some capacity. Here's an example from Columbia University. On this page, they specifically state appointment types that allow for PI rights. If you are not in one of these appointments, you do not have the authority to seek funding. Universities typically have "exceptions", which involve filing paperwork. You typically need a PI who will sponsor you. Columbia explains their policy on this on this page as well.
The idea here is that you are not in a position that is typically entitled to facility and administrative (F&A or "overhead") benefits, such as having a research administrator assigned to you. The school then has to make that resource available to you at some cost. You also would not fit into the typical mentoring circles that most PIs have access to, where they receive mentoring on technical aspects such as project management. A research administrator can't tell you how to fill out your progress report, only that it is due. Without a dedicated PI mentor, the university would be loathe to approve PI rights.
There are other concerns too:
- What does your budget look like? Are you going to need help procuring items? Someone to process reimbursements? Are you going to hire people?
- Is your appointment going to last the full length of the grant? Are you guaranteeing your entire salary? If not, does the university need to commit to the remainder of your salary for many future FY's? If so, who is that guarantee--the PI or a department budget?
- What happens if you leave the university? They don't have to let you take the money. They may take over your project and reappoint a new PI.
- What facilities do you need? Do you need a desk or access to specialized resources?
I will say that in the 8 years that I have submitted proposals, only once have I seen a postdoc get PI rights approved for a specific project. It took over a month to be approved by the dean's office, and it was because there was enthusiastic support from tenured faculty. I cannot fathom that anyone without a PhD could get approved for PI rights. It's not that there is always an explicit requirement, but there is a question of what category of researcher would be capable of being a PI and managing a scientific project but also does not have a PhD. Even if such a person existed, if their research and work was so compelling, a PI would surely vouch for them and enthusiastically mentor them. If not, they aren't worth the risk to the institution.
Being a PI is rough. It's really hard to learn how to procure and manage a grant even when you have access to the best resources. You need to work on getting the support from your advisers (and they turn your idea into a grant, but they are the PIs and you could be named specifically in the grant instead of employing generic personnel) or return to a formal path to PhD if you are interested in doing this type of work and having your name as the PI.
Regardless of your academic interests, at the end of the day, your institution has rules in place in order to protect it from individuals who may be unable to fulfill their commitments. Your current academic appointment suggests that you cannot continue this work unsupervised, so you need to remedy that first.