So, I figured out the problem - in order to actually run this without RPG Maker XP installed on your system, you have to have the RPGXP RTP installed on your system already. Here's a link to where you can get the RTP: http://www.rpgmakerweb.com/download/additional/run....
I have no idea currently if there's any way to include the game and the RTP in the same download (I'm sure there is), but this can work as a workaround until I can find a better solution. I also changed the download from the .exe file to the .zip file. Let me know if that works any better, and thank you for alerting me to this problem!