Improve search in web documentation

### Describe the project you are working on

Small game projects and a pixel drawing app.

### Describe the problem or limitation you are having in your project

Finding things in the online documentation has been difficult for me because the search functionality is so limited. I often have to resort to external search engines to find if something has built-in support or not. Even finding things I already know exist is difficult; searching for "CharacterBody" doesn't find "CharacterBody2D".

With the current state of the search I feel bad directing people to look for answers in the docs, when I myself have a hard time doing it.

### Describe the feature / enhancement and how it helps to overcome the problem or limitation

My current idea is to use [Sphinx's Web Support](https://www.sphinx-doc.org/en/master/usage/advanced/websupport/quickstart.html). If I've understood things correctly, it builds search indexes when the docs are built. I haven't tried it yet but it sounds like it could be a big improvement.

I'm not sure if it's the best way, I'd love to hear if anyone has other suggestions. If this proposal is accepted I can implement it.

### Describe how your proposal will work, with code, pseudo-code, mock-ups, and/or diagrams

I would add the `sphinxcontrib.websupport` extension and set it up in `conf.py`. Then add Python functionality under `_extensions` to use Web Support when the search is used. I haven't worked with Sphinx before, but I'm guessing that's how it'd go.

### If this enhancement will not be used often, can it be worked around with a few lines of script?

No.

### Is there a reason why this should be core and not an add-on in the asset library?

This is about the docs website, not really about the engine. I'm not sure if the proposals repository is used for the website too but I thought this one might warrant it. Either way, it's big enough that I feel like I need feedback before I start implementing it.