Of course they are. The vaccines met FDA's standards for safety, effectiveness, and manufacturing quality needed to support emergency use authorization.
I don't know who or what started the rumor that the FDA did not give these vaccines approval for use. They did.
The currently available COVID-19 mRNA vaccines have been reviewed and authorized for use by the US FDA, under an Emergency Use Authorization (EUA) process, and so technically they are not considered fully approved. This EUA process is used in times of urgency to enable medications to become available more quickly than usual. This had never been used before for a vaccine