Definitely ask before you GO to Disney. Whatever you do, don't wait to ask until you get back. If you leave the stitches in too long, you get the "railroad track" scar because the skin starts to form scar tissue around the individual stitches. I have some nice specimens on my legs from some mole removals, and you don't want those on your daughter's face, so keep up on when the stitches need to come out.
I know no one wants to interrupt their Disney vacation to "reenter the real world" even temporarily, but this could leave a lifetime scar if you don't. The urgent care place with the shuttle sounds like a perfect solution.