You can get several helicopters anytime, but you need to have access to the Saints HQ Penthouse to store them. Outside the penthouse, past the pool, there is a helipad. Take the helicopter there, then once you land on it, press E (or whichever key) to store it. There are two different helicopters at the airport, along the west building (along the water). On the army base island, in the middle yard with 4 helipads, there are 7 Eagles and 1 Eagle with miniguns and rockets. The weapon variant Eagle is on the southeast helipad. Also on the army base island, on the helipad on the roof of the building, there is a Tornado (so you can just store that one, instead of buying heli delivery).
Once STAG arrives in Steelport, you can get a few VTOL aircraft. Althought they fly like a plane (and an alternate hover setting) you can still store and keep them in the Saints HQ penthouse helipad.