Firstly, recognize that they probably placed that ToS there for some or all of the following reasons:
- They want a legal basis for suing someone who tries to steal stuff from their site.
- They want a legal basis for suing some idiot programmer who brings down their site because of a poorly programmed bot he wrote "for a project".
- Their lawyer told them to put these ToS up because he saw them on some other large site and thought they looked good there.
You likely fall in the second category (and maybe the fourth, I don't know what other things they're protecting themselves from). To that extent, you have to convince them that you aren't an idiot who will run amok on their servers. If you're aligned with a well-known university then maybe you'll have more luck, but even then it'll be a stretch... this is a for-profit company, not a research institution. Even if you can convince them that you're not a total idiot, they still have very little reason to want to grant you this permission. Your pet project will gain them nothing, and may in fact demonstrate something bad about their site that they don't want public. The "I'll share my findings" argument really doesn't go very far... companies have their own interests, governed by many things, including politics, agendas, egos, and what have you. Your powerpoint presentation that may be done in a year or two that will probably be too technical for a business VP to understand anyways won't give them any business value.
The point of all the preceding words is just to convince you that you have quite the uphill battle here.
All that said, I would just start calling every "contact us" number I can get my hands on. Definitely start with the numbers on the "Contact Us" page. Don't be afraid to send a written letter! For phone calls, I like the phrase Jack St Claire used in his comment: "I have a weird question."
Success is easy to determine. Check your mail... do you have a letter (written or electronic) from the TripAdvisor legal team permitting you to perform the research? If no, then you haven't succeeded yet. Keep trying! Perseverance will be key here.