There are a lot of security plugins for WordPress, and they can sometimes cause problems for other plugins by blocking requests.
If you are using a security plugin such as Wordfence and find that Search Regex is showing errors when making requests then you can try temporarily disabling your security plugin.
If Search Regex starts working then you will know that the security plugin is blocking requests. You will then need to configure the security plugin to allow Search Regex requests through.
Search Regex uses the default WordPress REST API.