Your institution (and specifically at the school level if you are organized into schools) will have rules about who can submit applications as a PI. It is generally accepted by NSF that if you have PI rights you can submit to their call as a PI, unless the solicitation requires you to hold a specific position or other qualifications. You should check with the chair of your department to find out who gives those rights out and what the process is to obtain them. You probably need a tenured-faculty sponsor (internally, not necessarily on the grant proposal). I work at a prominent US research university, and the "exit strategy" is always a calculation in these conversations. "What if you leave?" is always going to be a question. If your answer is, "I will leave and will take this with me," the decision becomes murky for your Dean. They have administrative costs to propose this grant, accept it, manage it, and transfer it. What is in it for them? If you aren't intending to leave, but must admit it could happen, be ready for an explanation as to what will happen to the grant and the people attached to it. You should be prepared to submit a SOW (statement of work) along with the requisite form(s).
In terms of the process of transferring a grant; while the decision to transfer rests with the institution that receives the award (not the PI), it is generally considered good sportsmanship in academia to always transfer grants when PIs leave, unless the grant is highly collaborative or a large part of the research team/facilities remain. At that point, a subcontract at no additional cost may be issued to the new institution (which will cost extra dollars in indirect costs). In your case however, it may be that if you leave, your institution would look for a new PI rather than transfer with you. You really will never know though until that day comes--it's on you to convince them to transfer it.
Your first step is to procure PI rights. Do not delay on this. If approved, find out immediately who will help you submit this grant. You need to be connected to a research administrator ASAP to ensure all your documents are in order. Do you have a Research.gov ID yet? Have you read the NSF PAPPG to get documents together? These things should be worked out sooner rather than later.
Be advised: If you are submitting after 10/1, new forms will be required for the biosketch and current and pending, and I cannot stress enough the value in you checking in with an experienced administrator as soon as possible to find out the details of this process at your institution. It is very regulated and stressful when done last minute (within 10 days prior to the due date), even for very experienced folks. The staff will almost certainly have multiple submissions, and you want to grab their time when they are more free and less stressed out. The new regulations are already causing people grief, and the fall is the busiest time of the year.