My personal Apple ID is not the same as my company Apple ID. Do I just need to log out of the App Store with my personal ID in order to download El Capitan?
Download via Mac App Store with Different Apple IDs?
That appears to have done it. You can sign out of the Mac App Store with your personal ID and sign back in with your company ID, without having to do so system-wide.
Don't worry about credit card, just paste your redemption code into the provided input field.