How to handle PUT and DELETE HTTP request in a PHP API First App?

You can absolutely also do PUT requests from browsers, but you need javascript.

Generally I would say a good way to think about it, is as follows:

First build a great REST api that follows all the rules. Only once you are at that point, think about the workarounds you need to make it work in other contexts. Submitting an HTML form is a valid thing to need a workaround for.

However, since 'POST' is completely open for interpretation, and has little rules associated, one option would be to create a single resource (or url) on your server that handles all the POST requests coming from browsers. (something like /browserpost).

You could always add a hidden <input> field with name="url" that specifies which resource was actually intended to be updated, and an <input> with name="method" value="PUT" for the intention.

You will need to add CSRF protection anyway, so I feel this would be a solid way to deal with this this. 1 endpoint to specifically 'proxy' html-based form submissions and internally do the appropriate mappings to the correct REST services.

I would use GET POST PUT DELETE as they are described in HTTP. That's restful (in my opinion). As regular browser forms does not support this I would send the data via AJAX.

If you really need to use browser forms, maybe because javascript is not enabled, then using POST requests with a param like ?method sounds like a suitable solution - although I don't like it.