I'm having the same issue with both the original game and the Forces of Corruption expansion.

Whenever I click the game icons, a system error pops up on the screen stating:

The program can't start because d3dx9_27.dll is missing from your computer. Try reinstalling the program to fix this problem.

I've tried reinstalling and it didn't help, I get the same error. I know this file has to do with DirectX, but my DirectX is up to date (DirectX 11). Has anyone else had this issue and were you able to solve it?

Thanks!

DX 11 doesn't contain most libraries for 9 and earlier, which is what you need. Here is the relevant support article:
https://www.gog.com/support/star_wars_empire_at_war_gold_pack/directx_file_not_found_d3dx9_43dll_etc

Follow these instructions and it should work. Or, at least, that particular problem should get resolved ;)

Thanks for the info, I will try it out.

Also, willl doing the above fix screw up my DirectX 11? Or is it just adding the libraries for the older DirectX?

Nope, it's perfectly compatible and required for virtually every game on GOG (or anywhere, for that matter).