Azazel wasn’t captured, he was killed.
There’s a newspaper article, The Bent Bullet Report, which was part of the promotional material for Days of Future Past. It’s an explanation of President Kennedy’s assassination, and the alleged role of mutants (namely Magneto) in his death. The full text includes a few mentions of Azazel.
Indeed, two members of Lehnsherr’s own Brotherhood of Mutants, an organization the FBI classified as “a pro-mutant terrorist group,” were killed [in 1963]. The mutants code-named Azazel and Tempest were slain by Project: WideAwake operatives in July.
There’s an explanation of Project: WideAwake elsewhere in the article:
Project: WideAwake was a covert CIA task force President Kennedy created after the Cuban Missile Crisis to investigate other X-Gene cases. While its mission was identification and research, it did have paramilitary autonomy. Records state Azazel and Tempest ambushed Project: WideAwake operatives, and died in the firefight that followed.
I’m sure Trask (or anybody) would have preferred Azazel alive for their studies, but it’s hard to be picky about these things. A stray bullet at the wrong time, and he’s dead, teleport or no teleport.
Remember that Azazel would probably be facing soldiers who had some idea of mutant capabilities – possibly even with specific intelligence about Azazel’s powers. Teleportation is a useful skill, but it doesn’t make you indestructible. If he was complacent against a group of operatives trained to handle a teleport, it’s very possible he’d be outmanoeuvred.
WideAwake is not a new concept; it’s part of the Earth-616 universe in the comics. This also isn’t the first time it’s been seen on screen; it’s a folder on Stryker’s computer in X2: