In Virginia, you must be 21 in order to own a handgun.
There are no exceptions.
If you are 18 with a handgun, you must be using it under the supervision of a responsible adult over the age of 21.
Now, there is a lot of media attention about the "Gun Show Loophole." It is misrepresented. If you have a table at a gun show and are selling firearms, you must have a Federal Firearms License, also called an FFL. Spending money for a table creates a "commercial action" thus you have to have an FFL to sell a firearm or you are committing a felony. However if the person at the table is just selling beef jerky then they do not have to have an FFL no matter how lethal that beef jerky may be.
Now, a private individual may sell a firearm to a third party without the need of an FFL. This is legal unless they are knowingly selling to a person who can not own a firearm. The Federal Courts have held that selling to an underage person is a criminal act because the seller must take actions to verify the age.
Person to person firearms sales are handled in many ways. It is true that a person can go to a Firearms Show and walk the isles with their firearms and advertise that they are for sale. They can also post firearms for sale in the classifieds, trade publications, and even online.
In Virginia, you do not need a "license" to own a firearm. You receive an instant check by the Virginia State Police through NCIC which is the same system that police officers use when they check a person whom they have stopped and need criminal histories. This is not a "lesser system" that has been argued by several opponents of the system. This check is the "real deal."
Thus, to answer your question. An 18 year old can not purchase a handgun at a Gun Show. However, they can purchase a long gun. IE: Shotgun or rifle. Also, an 18 year old can not purchase handgun ammunition. However, this is a somewhat archaic law due to the fact that many long guns fire "handgun" ammunition and also there are several handguns which can fire long gun ammunition.
I hope that this helps.