I've worked in both large and small. In both I've had the experience of administrative folks holding up the works (intentionally or otherwise). The biggest difference for me - in my small school they were in the same building or right next door, and I could go bug them in their office when necessary. They also knew my name and my face. Once the finance clerk left early on a Friday without signing off on the PO (which had been on her desk since Monday), when I stopped by to ask about it her boss told me she was gone but went and approved it right then without even looking at anything, because he knew if I was asking for it then it was legit. In my current position the finance folks are a 25 minute drive away, in a building I have no other reason to visit. I've been here 2 years and haven't met any of them in person. I haven't gotten desperate enough to drive over there, but sometimes it takes several rounds of emails and phone calls to get things pushed through.

In my experience, as long as you're doing your part of the paperwork, there's no disciplinary action coming your way because stuff isn't done. I wouldn't stress about that.