I'm not sure what I did or how, but it seems to be working now. Yesterday I tried running it in administrator mode, which didn't seem to change anything so I just turned off my computer and decided to mess with it later. Though this morning when I tried to trigger the error with Process Manager running, the error just wouldn't trigger and it let me do everything normally.
I'm a little worried the error might happen again if it is some sort of hard drive error so I'll keep an eye out for that. I've had errors with my external hard drive in the past, even alongside this error (specifically trying to move different Steam games between the internal hard drive and external hard drive using the "move install folder" in the Steam properties tab, giving some error like "disk write error". Ditto with trying to install programs to the external hard drive) so that could be the cause of it. I'm really not sure..
My installation path for the program itself was C:\Program Files (x86)\Steam\steamapps\common\RPG Maker 2003 and the project file path was C:\Users\[name]\Documents\rpgmaker ... However, while trying to fix the error I found I had the program installed on both of my hard drives, I'm not sure why. I had multiple project file paths as well, which seemed to be fine before but I ended up deleting the program file folders on the external drive as well as all of my project file folders on both drives just in case. I could still provide a log file if needed but I don't know how helpful it would be considering it's working fine now, though I could in the future if I ever have the issue again.
Thank you all so much for the help. I still have no idea what caused it to work but I'm very grateful either way. My apologies for the "steam/stream" error, I didn't realize it said "error writing to stream" either until Andar pointed it out lol. Thank you again.