You've been given lots of good suggestions, and I just want to echo the idea put forth by a couple of posters that the 14-year old should be an active participant in your conversations. After all, they are her anxieties, ones I'm sure she'd rather not have, and she's old enough to accept certain responsibilities. When raising the idea of medication and other coping strategies, it would be useful to hear her ideas. Indeed, if she's encouraged to take responsibility for the situation--for instance, doing some research (alongside her parents, of course) on medication, mental activities, aversion therapy--might very well have the happy side effect of helping her become an advocate for her own health in the long run. And, of course, she needs to be willing to implement whatever strategies are agreed upon if they are to be effective.
I applaud you for being willing to work with her in this situation. It sets a good example for your DD on how to behave toward others with compassion and patience. And it also sets a good example for the kinds of behavior you would expect DD to have if the situations were reversed and she were to be a guest of someone else...
Finally, to be optimistic, I can't think of a better incentive than to face one's fear of flying than to go to WDW with a friend. (In fact, I think that facing one's fear with a supportive group of people who aren't one's family--sort of neutral observers--might work out very well for your DD's friend.)