Where does your starting gear go in The Pitt?
When I arrived, there was a safe at the front gate, but when I came back, it was gone.
Is all of my gear gone as well?

It's stripped from you, you're a slave not a rich merchant.

And yes, it'll be given back a bit later. Follow the main quest, once you whip enough behind in the pit matches you get all your goodies back.

Cool - but I have completed everything and get the "retreive your gear" sub-quest, but it is no where to be found - is there a trigger to its return or do I have to find it somewhere?

It sincerely astounds me how so many people miss to get their items back from the box in the arena room.

And there's an intermittent bug wherein if you don't get the items when you first get the option to, the footlocker disappears. Never happened to me, I only missed my gear the first time and spent time looking for where they hid it until I went back to the arena room.