GOG guarantees the games will run, but any extras like map editors, unless they're included in the game card, might not work out of the box. This is especially true for older games.

I don't have HoMM2 from GOG, but here is how you can try running the map editor.

First of all, I assume you got the DOS version of the game, which is the regular one. If I'm not mistaken there is the Win version under Extras that's not officially supported, so you can try that first and see if that one comes with a Win version of the map editor.

If that doesn't work, do the following.

Right click on the game shortcut and click Properties. Under "Target" you should find a reference to a .conf file.
Find that .conf file and make a copy of it. Call it editor.conf for example.
Open editor.conf in notepad. Look under Autoexec section which should be somewhere at the bottom. Change the game exe to the editdor exe.
Make a copy of the shortcut, then change its Target to refer to the new .conf file.
You should now have a shortcut to the editor.

Alternatively download and install DOSBox
https://www.dosbox.com/
And use it to run the executable.

Note: All this assumes the editor that comes with the game is a DOS executable which I'm not 100% sure. If it's actually a windows executable, DOSBox will not work.
Can you tell me what message you get when you try to run the exe?