Creating a vibration device based on bass in audio
Deciding on the actuator will precede any decision on the driver. 

Philips AmbX many years ago did a rumbler for gaming, and XBox controllers have rumblers built in that consist of nothing more than small motors with eccentrics attached - just a bigger version of the vibrator in cellphones. These can be made to rumble at varying frequencies depending on drive voltage (most are just brushed DC motors), but are pretty crude and not great for syncing to music though acceptable for sound effects.

For a better relationship to the actual bass you need a device that can be made to  have a displacement that follows the bass, at last over a limited frequency range. A mass attached to a linear motor or solenoid is an obvious choice - looks like that is what the Basslet does, but probably the cheapest way is to use a speaker. I have at home one small Bluetooth speaker that really thumps a lot more than the larger versions I have of the same, and the trick is the use of an auxiliary bass radiator. This is similar to a speaker, without the voice coil, and with additional mass attached to the cone or plane, which gives it a far lower resonant frequency than a similarly sized speaker could - at the expense of a limited range over which it can resonate. The surprising thing is that it can be driven by the pressure generated in a sealed enclosure by a bass driver only about 60mm in diameter.



This is one I found on Alibaba that is similar. This then becomes more of a mechanical project, with a fairly simple amplifier that would need a low-pass filter. The ABR can be tuned by adding mass to lower the resonance.