I swapped out of Music to Chemistry nine weeks in to the course and am in the process of catching up basically most of the first unit. Head of Sixth wasn't keen on the idea but I managed to talk my way into being allowed as Head of Chem was happy for me to do it (he taught me in Y11 so he knows me). I'm getting Cs in it currently, which although it's better than most of the class it's way below my grades in my other subjects and I have had to work harder than I ever have before to get this (although I was really lazy at GCSE and up until this point at AS so that might not mean anything).
I don't take sociology so I can't tell you how much you'd need to catch up on, or how hard it would be, but I think the only reason I got away with changing so late into the course is because I'm a high achiever and known among the staff as someone who learns easily. Also, I'd already covered a lot of the more basic AS Chem content in triple science GCSE.
I think that if you really aren't happy in Art you should drop it regardless, as the stress from it may bring your other AS grades down and that would be utterly pointless (and besides, your mental health matters). If you've come to the realisation that Sociology is definitely something you would LOVE studying and putting in all the work to catch up would be fun for you, then go and ask again. Looking at the spec for your exam board is a must, find out what you need to know and learn at least some of it. Talk to the sociology department/other students who take it and find out exactly what you've missed. Revision videos can be helpful although they're not a substitute for proper lessons/textbooks.
If they still won't let you do it, you still have some options if you really want to do sociology:
1) If your circumstances allow it, maybe you could get a tutor to teach you?
2) Try and learn it yourself - this isn't such a great idea as you need a teacher really
3) Drop Art, use your extra frees to study for your other AS levels (so you do really well and definitely won't need to do any resitting next year) and then take up AS sociology next year while you're doing your A2s.
Sorry for the essay, but ultimately it depends on a) if they will let you do it (you should still ask anyway, and argue your case as well as you can, don't let them just tell you no straight off, maybe bring in a parent/guardian/supportive adult for moral support in front of intimidating senior staff) and b) whether you think you can cope with the workload on top of whatever else you're doing. Bear in mind that this is potentially the rest of your life you will be gambling on, and as such if you decide to do it it's imperative that you know what you're letting yourself in for. Having said that though, unless you're aiming for a top uni (which I presume you aren't as they don't think the -ologies are rigorous enough) it doesn't matter if you have to resit things/don't do as well as you might have hoped/do a 3rd year at sixth form, you can still go on to do what you want. So I think you should go for it if you want it badly enough.
With regard to your question about paperwork, I didn't have to fill any in at all, it was almost ludicrously simple once I'd got the all-clear to do it. It took about a day to get put on the registers and get my timetable updated but that was about the only admin-type thing that had to be done.