It’s very easy enough to have images point to a page on your site, but keeping track of the hunters is a challenge.
One way to do it is to capture the user’s IP number and save it. Then the link on the other site would look like this:
<a href="URL of your page?site_id=1234"><img src="path to image"></a>
Where each of the remote sites has a unique ID. On the page at your site, you’d capture the IP again, check the site_id argument and give that hunter credit for the find. You’d want your own custom database table with the hunter’s IDs, names, and site_ids they’ve successfully found the token on. You could have a field for each site_id, or put them all in one field, separated by commas -- adding the new find each time they come back to your site if it’s not there already.
It’s not all that complicated, but the many users with dynamic IPs would have to do the whole hunt in one session. A better alternative might be to save a cookie on the user’s machine and use that to ID the users on return visits. If they clear cookies or have them turned off, though, they’re out of luck.
It would be pretty easy for people to cheat if the site_ids become public (and they’ll be visible to all users in the URL back to your site). They could get credit for the finds without visiting the other sites at all just by entering the right URLs. If that’s an issue, the only totally reliable way to do it is to have the user sign in with a username and password on each site they visit and have code that validates them through your site before sending them back, but that’s probably not practical. It could probably be done without them signing in using JavaScript on each site, but the other sites would all have to install the script in addition to adding the link.