The dealer does not verify anything about the purchaser. The gun MUST ship from the online store to a registered FFL where the purchaser must show up in person to complete the paperwork for transferring the firearm, show identification, and submit to a background check. The FFL will call in the identification info and will find out immediately on the phone if the sale is allowed. If a long gun is purchased (rifle, shotgun) the purchaser can leave the FFL immediately with the gun. If a pistol is purchased, the pistol must be picked up after a waiting period which depends on the state - it is two days in my state. Guns are never transferred directly from an online store to the purchaser.
FYI: There are a number of states with no waiting period for pistols, including mine.