You can use an established company website, search online for render farms and see if they have an API to download.
Integrating the API in Blender is something you’ll need to do yourself though if you go that route.
That is what you’re asking right?
“I’m not quite looking for a render farm, I’m looking for a real-time API service that’s integrated into Blender as an extension.”
You want an Add-on (API service that’s integrated into Blender as an extension). Add-on or extension, call it whichever it’s basically the same thing on a user level.
The add-on UI is what the user will use to send the render to the render farm directly and when completed receive the completed render. Automated process.
Instead of saving the .blend file, uploading to a render farm website to render, then download the completed render. Manual process.
If your looking for faster rendering locally, there are add-ons that can do that, but all they do is tweak the settings in Blender and optimize your hardware. These tend to often save at the most a second per frame. IMO not really worth it unless your rendering a half hour animation.
Even then I would suggest rendering in batches rather than a days long render marathon.
Of course there is one more option…
Upgrade your hardware.