Ordinarily vaccine developement and FULL FDA approval does take years. During emergencies(pandemic) or for conditions that have no approved treatments available, the FDA allows other means to get vaccines/therapies to the public.

Emergency Use Authorization (EUA) is one option and is what has been used for covid to my knowledge. This is typically given after Phase 3 is complete. At that point the drug/vaccine can be marketed(sold) while the company continues following up with the trial(Phase 4). If Phase 4 is successful the FDA provides full approval.

https://www.fda.gov/vaccines-blood-biologics/...-explained