Looks like you're legal.§ 14‑269.7. Prohibitions on handguns for minors.
(a) Any minor who possesses or carries a handgun is guilty of a Class 2 misdemeanor.
(c) The following definitions apply in this section:
(2) Minor. – Any person under 18 years of age. (1993, c. 259, s. 1; 1994, Ex. Sess., c. 14, s. 5; 1993 (Reg. Sess., 1994), c. 597, s. 1.)
As for carrying proof that you bought the handgun in Virginia, you don't need it. You don't have to prove you bought it legally, the state has to prove you bought it illegally. Besides, there's nothing that says an 18-20 year old can't get a pistol purchase permit.