As an electrical engineer currently in the process of getting a dental implant, I would say it's definitely doable. But it would present a pretty serious packaging challenge, particularly the power supply.
[UPDATE] Turns out you can get dental implant hardware on eBay:
In 1945 the Soviet Union gifted a wall hung artwork to the USA embassy which passively transmitted audio signal from the room with no power supply or active electronics.
No internal power supply. Very important distinction.
"The device, a passive cavity resonator, became active only when a radio signal of the correct frequency was sent to the device from an external transmitter."
But the Thing was a transmitter. A receiver could be entirely passive and very small. It could easily fit in a dental implant. (There have been reports of people hearing AM radio broadcasts through their fillings.) But the risk of detection would be very high.
It's a pretty straightforward calculation, which I don't feel like doing right now. But I think Stockfish is going to be pretty power-hungry at the grandmaster level.
I have some smart light switches in my house that don’t contain batteries and aren’t wired to mains. They’re powered purely by the force of pressing the switch. A bite-powered receiver might not even need batteries. Okay, maybe I’m getting too crazy?
[UPDATE] Turns out you can get dental implant hardware on eBay:
https://www.ebay.com/itm/272267166511