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Maksym, you must be an Asylee (not pending). An Asylee is not a non-immigrant thus the federal law permits the purchasing and owning of a gun. However, one should check the laws at state level . The only issue is you will run into gun stores that are not very knowledgeable with firearm laws and will deny you the purchase unless you show them a "Green Card" - regardless of whether FBI approves you or not.
I would check with a Local FFL to be 100% sure
Maksym, If you have a valid Florida drivers license. I would walk into a local gun shop and see if I could buy a firearm. That will answer your question and you will have developed the relationship with a local FFL to use for transfers to buy firearms online.