There was a similar question to this a few weeks ago. My answer was (condensed)
The general consensus is that it is safe to fly when pregnant up to about 32 wks for international flights and 36 for domestic. Though this may be more due to airline's fears of emergencies in the air than mother and child safety.
THey are all outlined in this site
http://pregnanttraveler.com/Articles/Air%20travel.htm
A doctor's letter may help but you need to have clear communication with your airline to get them to confirm they will be happy to carry her home at the end of the holiday. You are just on the brink, at 32 weeks. I don't think there are technically any immigration rules for the US barring entry to pregnant women; though if you were coming in at 40 wks they would probably be a bit suspicious ...