This is how I set up Vite to meet my requirements for building a Web Extension for Google Chrome, since other Browsers also support Web Extensions with Manifest v3 it should also apply to Firefox, Edge and Safari.
To build the project, first install the dependencies with npm i and run npm run build to build the extension. If you are using Google Chrome you can then load the extension in Settings/Extensions by clicking on "Load unpacked" and selecting the dist folder of this project.
Currently the package @userbrain/ui is not published to npm, so you will have to clone the repository and run npm link in the root folder of the package and npm link @userbrain/ui in the root folder of this project.