Yes, it will be a problem in that you'll miss most if not all ASWs, and that any reimbursement they offer will be much less enticing. Just make sure you have Internet access and someone willing to scan you your mail, and the applications themselves will be fine.
I ended up taking a(n expensive) trip back for a couple weeks to go to 4 ASWs and those schools pushed the others out of consideration because I had personal experience with them. I don't think I would have been terribly unhappy with any of my choices, but I'm definitely happy I got those ASWs in.