I often refer to the Seattle Regional as "the end of the robotics season.... unless we do something silly and win."
As much as a challenge as it would be, though, to have to come up with that kind of cash on short notice, it is still a goal we aim for every year.
First of all, if you've got a sponsor supporting your airfare, you're probably already ahead of a lot of teams. That is really great.
A lot of the fundraising, though, seems to come if you can get the school and community interested in supporting your success, and letting them know what you need. You need to find a way to get the local media to latch on to your story. We had an FTC team at a neighbouring school a few years ago that was so successful with their fundraising to go to Atlanta that they built up enough momentum to keep the support going well into the following VEX competition season. You might be able to leverage this into new supporters and new sponsorship on a continuing basis for your team.
It is also possible that parents on your team will want to kick in to help with travel costs. We normally charge students a nominal fee to attend a regional (but never leave a kid who can't pay behind), and have, on occasion, had parents step forward with very generous donations, knowing that their support would help not just their son or daughter, but the whole team.
But media coverage is key. Contact everyone... from your city council, to chamber of commerce to... well... everyone. Make some noise about your win... and your opportunity.
And you might find some new supporters almost as awesome as your sponsor who's chipping in for the flights!
Jason