I had similar issues than once when I was a student, but I was always able to get my key at Public Safety. I don't know the particulars of your case, so maybe there was some reason why that was not possible this time.
I would advise you to do a little more digging and try to figure out what exactly the problem was. Tell ResLife what happened and ask them what you should next time to ensure you don't have to sleep in the lounge. And don't accept "you have to get here by X time" because that doesn't work. There has to be a plan B in case you miss the deadline. If you don't get satisfactory answers, go up the chain of command until you do.
If you find that there simply are not policies in place to accommodate people in your situation (a likely scenario since Drew has a serious policy deficiency), then I think the petition idea makes sense, and the specific goal should be to convince the administration to create such policies. If policies exist and were not followed in your case, then a different course of action is needed.