How do I buy the cart.p8 for my handheld?
Maybe through the Itch launcher? The launcher just downloads the web build, and some times the web build is just a wrapper for the actual game.
And the cart can be downloaded from the site of the engine's developer for sure:
https://www.lexaloffle.com/bbs/?pid=73825#p
Click on the pink and white "cart" (or the "cart" text besides it) on the bottom left of the web player and download the .p8.png image. That's the cart and is playable on the PICO-8 engine, so it possibly runs on emulators too.