Rotten Tomatoes required an API key, which meant I'd probably need to build some intermediate service. (I of course wouldn't want to include the API key in the Chrome extension.) I didn't want the extra overhead for that. I looked at other services that did have an API without authentication (like Trailer Addict), but their player required Flash, and lacked a bunch of content (like for TV shows).
YouTube has a better player interface, is more likely to contain obscure content (e.g., from old movies, or relatively unknown TV series), and had a super low barrier to entry, so I went with that.
I remembered poking at IMDB awhile back and that there wasn't an API, but I just found http://stackoverflow.com/questions/1966503/does-imdb-provide.... It seems that endpoint will give me the IMDB movie ID, which you need to show the trailer, so I may end up switching to this.