To begin, this is important: The bavarium wingsuit is a part of the Sky Fortress DLC. You cannot obtain it without the Sky Fortress DLC. This can be bought in the Just Cause 3 store page. Buying this unlocks the set of missions relating to Eden, the "Eden airship" regions (including special weapons that can be found there) and the Bavarium Wingsuit.

The first step to getting the wingsuit (ensuring you have the DLC installed and have succesfully loaded a new save etc.) is to complete the first few missions. Point is, you need to get to the mission named "A Terrible Reaction". The start point for this mission is in Sirocco Nord. Completing this will unlock the Sky Fortress missions.

Upon completion of "A Terrible Reaction", you should get a call from Sheldon, which begins a mission entitled "Suit Up". This is the mission which unlocks the Bavarium Wingsuit. To start this mission, look for the yellow bull-head marker on the map which is almost directly north of where the last mission ends, on the central island.

Fast travelling here will take you to a field with a crashed drone, and the bavarium wingsuit. This quest is essentially a training course for the wingsuit, and it can be tricky at first, but it become easier with time.

now that you have reached the sky fortress DLCs, you will permanently obtain the Bavarium Wingsuit after completion of the Suit Up quest. It can be put on and removed from the menu, and cannot be used in wingsuit challenges. However, it is used for the wingsuit courses on the Eden Airship itself. Have Fun!

-=-Make sure you have the Sky Fortress DLC-=-

- Complete all main missions up to and including "A Terrible Reaction"
- Go to the location Sheldon has marked with the mission marker
- Begin the quest "Suit Up" and complete for the Bavarium Wingsuit to be permanently unlocked.

Locations etc. outlined within longer guide