Short answer: Order from the Stripe Terminal shop in your Stripe Dashboard. Anything sold there works with the app. Readers from places like Amazon won't pair.
You can order a card reader from the Stripe Terminal shop here: dashboard.stripe.com/terminal/shop. Anything you buy there is guaranteed to work with Payment for Stripe.
For most US merchants we recommend the Stripe Reader M2 (pocket-sized Bluetooth). For countertop use there's the BBPOS WisePOS E and the Stripe Reader T600 with its 8-inch customer screen. The Verifone V660p is a portable smart reader with a built-in receipt printer that docks for the counter. The all-in-one Stripe Reader S700/S710 runs the app right on the device, no phone needed. Outside the US, the shop shows the readers Stripe sells in your country.
One tip: always order through Stripe at the link above. Readers from places like Amazon are locked to other platforms and won't pair with the app.