Hi @simplythebest, thanks for your query. If you have Windows 10 Fall Creators Update, then eye tracking in natively supported. Using Windows 10 built in virtual keyboard you can use this to type and navigate the internet manually.
I am unaware of a virtual keyboard who output is directly tied to a search engine however. I did find a Chrome Extension that provides a keyboard within the Google Pages (Search, gmail, etc) @ https://chrome.google.com/webstore/detail/google-input-tools/mclkkofklkfljcocdinagocijmpgbhab
Maybe in conjunction with the eye tracking control suite, this might be a solution?