Your question is off-topic, but ... if you are only looking for a place to start, you might find "RTC Portal X" source code useful (see link below). Even though it has been
discontinued last year (there are NO WARRANTIES AND
NO SUPPORT) and the version released on GitHub does NOT contain Example Projects using FireMonkey (
only VCL), you should be able to find a cross-platform version of video encoder/decoder, as well as basic screen capture functions in its "Lib" folder:
https://rtc.teppi.net/rtc-portal-discontinued-and-released-on-github/