What are the issues to look for if a 2002 vw tdi beetle is ran without oil?

If you haven't already done so, contact the shop that left the plug loose and fill them in on the chain of events. Tell them you be taking to a shop to be inspected. It's always best to start communications as early as possible. No shop likes to get surprise phone calls late in the game. It can limit their options in finding a resolution that is fair to you and them. Then take the vehicle to a shop that has experience with VW TDI engines. Have them diagnose and document what they find wrong and take that info back to the original shop to come up with a resolution. The fact that you drove vehicle till it stopped running isn't good. Very likely major internal engine damage has occurred. I wouldn't drive the vehicle anymore and tow it to who ever is going to inspect it. Make you keep notes on when, who you talk too and what was said at the original shop.