Implementing this will require changes to both the search plugin and the theme I think. Fundamentally we would need to build separate search_index.json data files by language and then load the appropriate data and associated lunr files based on the active locale. These files are all loaded with each page which makes this seem possible. The path of the search_index.json file is embedded in minified javascript though, so that pushes things back towards "not simple".

This is a fundamental problem that came up during the accessibility evaluation. Due to how the search results display, there's no signal to alert a screen reader user that content on the page has changed as they type their query. Consequently, when they hit 'enter' or 'return' to issue their query, the site interprets that as a command to go to the first search result in the list, which, in many cases, causes you to end up on a page in a language other than the one you were using.

T307141: Multi-lingual search is not consistently multi-lingual is related and might imply that some locales would not have any search at all once this task is resolved.