Simply because sometimes things are too easy to be obvious.
Prescott Starbase
You wanna make sure to start at Union System’s Prescott Starbase close to the city center.
How to get started with the first vault
Start with a weapon preferably non-aoe (not your flak cannon) to avoid friendly fire issues. 
Get 50 m close to the terminal console and hack it. 
Next to the terminal you successfully hacked shoot the red lights as they appear (just one shot them like at a shooting booth) and the bars above (red/blue) change. 
After a couple of shots you complete the quest by flying close to the wall and loot the vaults items. (On screen it might look like you loot the actually wall not items in some cases, guess it is a bug. Don’t try to find some other entrance but if you do it should be on the back side with some boxes.) 
It’s as simple as that.
Thanks to Kito1Q84 for his excellent guide, all credits belong to his effort. if this guide helps you, please support and rate it via Steam Community. enjoy the game.
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