He was born in Syria. Was he a citizen
Doesn’t have to be a citizen, perm residence is all you need for a 4472 form and refugees are all nearly 100% given perm residence as a matter of course in being a refuge we NEVER send them home. So likely he was a green card holder and with no prior felony or domestic violence convictions able to pass form 4472.
My B he was a citizen
“Details about Alissa were slowly emerging on Tuesday. He is a naturalized U.S. citizen from Syria, according to two U.S. law enforcement officials briefed on the matter.”
According to the ATF, non-citizens can purchase firearms.
https://www.atf.gov/firearms/qa/may-aliens-legally-united-states-purchase-firearms