I'm currently working on a project that uses Mach3 or Mach4 to drive a CNC machine. Instead of writing the G-code by hand, I'll create software that does it for me.

Can I send my code directly from my software and have it execute without having to manually copy it into Mach3/Mach4?

Yes, mach3 exposes a COM interface and mach4 you can access the api's:

https://www.machsupport.com/forum/index.php?topic=34168.0

Mach 4 has a developers version you can get from them to do what you are wanting to do.
We used it to make our software run Mach 4.