So the game runs and I can play the tutorial missions but anything outside of that I am being told to enter my CD. My computer has no optical drive so I'm wondering if that is causing an issue somewhere even thought it's technically not needed.

I'm running Windows 8.1

strange, that shouldn't happen at all. I can't reproduce the problem, so all I can suggest atm is try reinstalling the game.
the missing optical drive is not causing the issue, since all gamefiles are in the gamefolder. i dont have one either and all GoG games run just fine.

I had this problem as well. I could still load save files but I could not start new missions anymore.
I deleted all the files for Settlers IV in the gog.com folder, and redownloaded the game. From the installer, I could click launch game when it was finished. Doing this allowed me to start the first mission again (all my save files were lost but I still have them as a backup somewhere else). Content, I quit the game. Later I tried starting it again, this time with the launcher from my desktop, and the same problem happened. So I have good reasons to believe this has to do with the launcher rather than the game itself. Strange, because earlier I was able to play this game. I'm gonna try some more stuff, when I find something that works, I'll let you know.

Right-click, Run as admin.

I think I fixed this by using S4_Main.exe in the /exe folder instead of S4.exe