Depends on the firmware of the device. Latest firmware (anything after version 5.18.5.01 - which released in October 2025) is currently not jailbreakable.

Jailbreak of any firmware after version 5.16.2.1.1 (June 2023) requires the Kindle to be connected and registered.

Anything prior to, and including this version, can be jailbroken with no registration.