Mozilla has the Oxidation project:
The goal of Oxidation is to make it easier and more pleasant to use Rust in Firefox, and correspondingly to increase the amount of Rust code in Firefox.
But it does not mention anything about whether Rust is supported for add-ons or not.
I don't think it is possible to write a complete add-on in Rust. Although you could use Native Messaging to communicate from the add-on to a binary built with Rust. When an add-on uses native messaging the browser starts a native binary on the system and exchanges messages encoded in JSON over stdio.
Mozilla has detailed information about native messaging on MDN.