Same ground truth as the Fire tablet: there's no iOS-style jailbreak for Fire OS. What you're actually going for falls into two tiers depending on your model and goals: Custom Launcher + Debloat, System-Level ADB Exploit, & True Root / Bootloader Unlock.

First, a terminology reality check: Fire tablets run Fire OS (a heavily modified Android fork). There's no "jailbreak" in the iOS sense. What you're actually looking for falls into three tiers, depending on your model and goals, Fire Toolbox, System-Level ADB Exploit, & True Root / Unlocked Bootloader.