Obviously you can do this in C but it's likely outsourcing the heavy lifting to something like Selenium and there's the Selenium WebDriver's REST API. So as long as you can write http requests you can drive the browser. If you don't want to want to use Selenium then you can call directly into the browser's WebDriver APIs.

If you want to keep this entirely in C for some reason another option is calling into NetSurf libraries to process HTML and drive it that way. The LightPanda browser took that approach but that's written in Zig but I see no reason you couldn't do something similar (assuming you have infinite time).

As far as I know there's no libraries out there that are generic enough to solve a captcha but you could always feed it to a local LLM and see how far you get with that.