I've been on FIRST event planning committees before, and I can honestly say to host a FIRST event isn't cheap! There are a lot of expenses involved, including many that the general public never sees but are necessary for the event to be held. Within registration fees, FIRST cannot absorb all the costs needed to run a competition.
Seeing how the registration fees are needed to run the event, it is possible to raise enough money in a short period of time to fund going to the Championships, all that you really need is to REALLY WANT to do it.
Where there is a will, there is a way!
In addition to the ideas already presented in this thread, here is another idea that can work. Look closely at the costs needed, and see how much of it the team really
needs to spend. The bare minimum needed is the registration fees; while funding the students travel expenses is a nice bonus, if you're broke you only need to focus on needs, and not wants. Have the students and mentors pay for their own hotel and airfare; this will usually cost about $400 per student if you can find a good airfare deal.
So now all you really need is the $5000 registration fee. In tight situation, if you feel that you can raise enough money in fundraisers to pay off the trip, have a team mentor take out a new credit card for the team and put the $5000 registration fee on that. (A six-month, no interest one is usually the best choice.)
Now you have six months to raise $5000. From our experiences, car washes can bring in about $500 in profits. Pasta dinners can bring in close to $1000. Pancake breakfasts can also bring in a lot. Can and Bottle drives (if your state has beverage container deposit programs) can bring in $500-$750. If your team takes the initiative and plans one or two large fundraisers like this each month, you can be sure you'll be able to raise the $5000 needed for the event.