Unfortunately, the browser extension is proprietary. They used to have an open source one but they stopped maintaining it.
Proprietary was a dealbreaker for me. There is no way to verify that it isn't selling everything I type even if I do have it configured to point at a local server.
I'm also concerned that the extension may eventually no longer work against local servers as well.
https://github.com/languagetool-org/languagetool-browser-addon/issues/247
As an alternative, there is harper by wordpress: https://github.com/Automattic/harper
It is webassembly and runs entirely in your browser.
EDIT:
I will add that the rest of the languagetool ecosystem continues to work fine. Libreoffice now has a built in client, which you can point at your own hosted server. VSCode [1] also has their own languagetool extension. I use those and those work great. But in the browser I use ~~harper~~ nothing. I should probably install harper.
[1] Well, technically I use [code-oss]https://wiki.archlinux.org/title/Visual_Studio_Code), which gets the extension from https://open-vsx.org/