No your own app can not write to the lazarus message app. As all apps are designed to run equaly without the ide => in which case there is no msg window.
However, there is a way for easy additional output. Compile you app as console app (assuming you are on windows) it in the compiler options, linking tab: uncheck the "win 32 app -WG" checkbox)
Then you have a console window while your app runs (on linux, just start it from a terminal).
You can write to that console via debugln from unit LCLProc. There is also dbgs() to format selected structures, and values.