I don't know of a device that can measure it at home, but there are mail-in saliva cortisol kits from companies like Genova Diagnostics, Labrix, and ZRT Labs. You usually send them several samples collected at specific times throughout the day.

They're mostly designed to be ordered by a healthcare professional, but you can get a few of them direct on Amazon, or through companies that order on your behalf (similar to healthlabs.com mentioned above).

I was given a big plastic jar with a certain kind of acid in it. Had to pee straight into the jar the whole day. Next day, the jar had to be handed over to the lab.