Alpha24, thank you for all of your time and help. We finally figured out our problem.
We had done the initial setup of the database, etc. through Powershell, and we set the sql mode to an empty string with a Powershell command, which meant that it worked fine as long as we didn't restart our server. However, we had to do a shutdown at one point, which reset our server, meaning the sql mode reverted to its default, but since the installation was already complete, it didn't give us any flags/warnings/error messages about the sql mode.
When we realized what the problem was, we went into the my.ini file and edited the sql mode properly, so it won't change again. Now it's working fine.