Yes, it is legal for a person over 18 but under 21 to buy a handgun from a private seller. Its up to the seller if they decide to sell it to you though. Make sure you tell them beforehand that you are "fill in age" and if they would still like to go through with the sale. If they decide to, do not show up looking like a "gangster".
Also before you OC for the first time, know the laws and the 1000 foot school zones around the area.
p.s. welcome to the site