Three keys to business: be likable, be flexible, be invaluable.
Always charge money for these types of integrations, especially with larger companies (that can afford it). If they can't afford a $5,000 integration fee, they aren't serious. Period. Move on. If you're likable, you can get them to commit something.
Once you get them to put some skin in the game (honestly, not that hard if your value proposition is right), be flexible. Make them as happy as possible without diluting your value (i.e. don't do a bunch of free work).
Don't worry about the 'They'll steal your idea' scenario. Big companies move slow. Small companies move fast. They need you just as much as you need them.
Look at any dev shop that has big clients; those clients have lots of development resources available internally, so why do they use outside consultants? Because some things take too long or cost too much to do internally. Focus on filling those niches and you become invaluable.
You'll find that lots of opportunities will open up with large clients. Look for them. Be patient and interesting things will happen.