The vast majority of advertisers on CJ use cookie based tracking.
The user's browser is set with a CJ cookie on clickthrough (with cookie duration varying per advertiser). When a purchase is made, a tracking pixel hits CJ with the purchase/conversion info and CJ matches up the conversion with the publisher. This is all done on only an hour or two delay.
Some advertisers do the tracking on their own servers and send CJ all the transactions in batch, usually once per day.
You can see which method they use if you look at the advertiser info on CJ. Tucked down on the left side is the setting:
