Hardware requirements for ambient active noise cancallation system using adaptive LMS
I would recommend checking out "Design of active noise control system with the TMS320 family" by Sen Kuo [[report link]].

TMS320 chips are specifically designed to perform various ANC tasks. In the report, they even provide code for programming LMS and FxLMS. 

In a lot of the literature on real-time ANC, the control system is implemented using these chip sets. So in my opinion, these chips would be a good place to start. Alternatively, I'm sure this system could be implemented with a variety of microcontrollers, but I am not aware of any other hardware specifically designed for ANC. 

Side note: I would consider looking into the leaky FxLMS algorithm as well. It's no more complex than regular FxLMS, but provides more stability to the adaptive algorithm (although reduces the steady state performance). The leakage term is typically necessary in real-time practical control systems.