It depends on the situation. If it is a hotel van with the hotel's name on it, I cannot imagine any DAL (or any other) crew trying to claim the van just for themselves. If they do they are jerks.
That said at many stations (JFK and LGA come to mind, but there are many others) we have specific PAID FOR vans that are "limo" services that are paid for by DAL, for DAL crews--just like many other airlines also have. Then you occasionally have freeloaders who think that just because they are airline crew (not necessarily on duty!) who think they have a right to a free ride into town.
Here are two good examples. We were staying in Manhattan and had a van just for us two pilots over to JFK. Another guy (in civvies) was waiting for the van and just hopped on board. He told me that he was in town for a few days and was a DAL pilot (he was, by the way). I just assumed he was on company business. It wasn't until halfway to JFK that he told us he was just in town to "have fun." The more he talked, the more evident it was that he was his own greatest fan.
The think that blew us away was his arrogance in not even requesting a free ride. Would we have been happy to do it. Absolutely! There was plenty of room and he was a fellow DAL pilot. We could not conceive of any other option. But he never asked--just took the free van ride because he felt he could. It would be like a jumpseater just hopping on board and saying, "hey captain, I'm on the jumpseat, I'll take a seat in the back, see ya later."
Even worse was a couple of years ago when I flew a charter flight to Cabo (yes, someone has to do it). We had this non-rev couple wondering if they could please hop on the crew van because they were a pilot/FA couple and they would sure appreciate it....turns out they weren't even DAL! The captain should have told them to go stuff themselves, but he didn't, so we were very cramped in a somewhat small van (the whole crew, pilots and FAs) so these freeloaders could avoid a taxi fee.
Make sure you get the whole story before you just bash the accused...and sometimes they too are wrong.