I can't cite specific examples because the ones I know about formed confidential business relationships, but I can say with confidence that this works All. The. Time.

That said, if you're some small-time researcher who can't offer a compelling business case to make this happen, then it won't be worth their time and they're likely to show you the door. [Note: my implication here is that it's not because you're small time, but it's because by the nature of your work you're not focusing on business drivers which are meaningful to the company/org you're propositioning].

Edit: Also be warned that if you're building a successful business on scraped personal info, you're begging to be served w/ a class action lawsuit (though take that well-salted, because IANAL and all that jazz).

Most of the time, in my (admittedly limited) experience. The two exceptions have been:

A giant site, who already had an API & had deliberately decided to not implement the API calls we wanted. I should add that another giant site have happily added API options for us. (For my client, really; not for me.)

An archaic site. The dev team was gone & the owners were just letting it trickle them revenue until it died -- they didn't even want to think about it anymore.

When the money is good enough :) . That is, usually not for startups, yes for established companies with money.

Never to my personal attempts