Is there a way to store extra ammo in your cargo bay so you can reload.

No.

Docking with capships lets you reload the same as docking with a station though, so once you build up high standing with various factions, keep an eye out for their capital ships. They normally don't let you dock with them if you aren't in the same mission as they are, but they do when you have good enough standing. It's a useful trick if you're behind enemy lines or in a hurry and don't want to return to the station.

No... but there ought to be.