Probably the easiest way to track purchases on a website you don’t control would be through an affiliate link/system.
If your partners use an affiliate system you can tag your links with the appropriate affiliate code. Most affiliate systems have panels for affiliate partners where you can track your stats.
Another option would be to try and setup cross-domain tracking on your Google Analytics account.
Alternatively, you can setup specific URL parameters and rely on your partners to pull out the relevant data.
These are just a few ideas of the top of my head.
Like I said, it all depends on the level of cooperation you get and your technical skills.