I'm an FFL in Florida. My customer is a FL resident and is currently in Utah. He wants to buy a rifle from me and have it shipped to Utah. Is this possible?

Perfectly legal under federal law.
Unless there is a FL or UT law prohibiting such a transfer.
The buyer would fill out a Form 4473 and the Utah dealer would run NICS on the buyer.

If you are asking can you ship directly to the buyer.........hell no.

Thanks dogtowntom. I know I can't ship it to him direct.

The hang up I'm having is that he is a FL resident staying in Utah. Because it's a long gun it's not a problem not being a Utah resident and doing the 4473 with a Utah FFL?

It's been legal since 1986 to sell a rifle or shotgun to a resident of another state, as long as the sale and transfer is legal in both states. (thanks to FOPA '86)
Prior to 1986 a dealer could only sell rifles or shotguns to contiguous states.

California for example prohibits the out of state transfer of any firearm to a California resident. They can pay for it, but the firearm must be shipped to a CA dealer.

Further, for the purposes of acquiring firearms, ATF considers you a resident of the state where you make your home. See ATF Ruling 2010-6 State of Residence

Thank you Sir. You are always a wealth of proper guidance & knowledge.