iPhone SE

Yes you can. If you know for sure they have AT&T you can buy it from AT&T or any place like Best Buy that sells AT&T branded iPhones. If you're not sure, buy an unlocked phone directly from Apple.

Make sure you buy it in the same country as the person you are giving it to lives, warranties don't transfer to other countries.

Yes! As you go through the online purchase process, you’ll see this screen (after having selected AT&T as the carrier, as well as having chosen the storage capacity, color, etc.)