There is a Prompt API in development that's available in both Chrome and Edge to give access to a local LLM. Chrome extensions have access to it and I believe websites can request access as part of an origin trial.

The model is fully managed by the browser. It's currently the Gemini Nano model on Chrome, and they are testing a version of the Gemma 3n model in beta channels. Edge uses phi-4-mini.

More information is available here: https://github.com/webmachinelearning/prompt-api