The 21yr law is federal and pertains to purchases through an FFL. Since private sales in Texas are legal (among Texas residents), an 18yo can privately purchase and own a handgun, barring any other conditions that would exclude him (felony, gang, etc.).
Well, if you ask them and they say no, then you're gtg. It's not your responsibility to prove they are telling the truth. Just don't be dumb about it lol