I actually use Omiture, so I am not 100% about how you would execute this with Google, but here are some options to check into.
First, if you know the page that your image is on, you could look at the links that users leave your site on. In Omniture, this is called an Exit Link and gives me the links that users click on to leave my site. If you can get this type of information from Google, you can probably segment it by page and from there, figure out how many instances there are of people going to the site you are linking to.
Another option is to pass some sort of parameter in the referring URL and have your sponsors track the number of times those parameters are referenced on their site. Not 100% how to do this on a development side, but you should be able to pass some sort of id on the referring URL with a '&rid=###' at the end and they should be able to process it on their end.
Yet another option is to have them install Google Analytics on their site, which is good for them anyway, and then set up the referral that way.
Ultimately, to do this, you are probably going to have to get them to do some work on their side, but hopefully, some of these ideas help.
Good luck!
