I really like Slack but my organization won't allow it if we cannot disable file sharing features. I'm trying to find out whether I can either disable uploads of certain file types or hide the file library all together. Their support says there's no way to do this natively but is this possible with the API?
No, not to my knowledge. The main focus of the API is not to configure your Slack team, but to enable bots and 3rd party apps to communicate with your Slack team and retrieve information. All core configuration for your Slack team has to be done in admin settings and apparently you have already learned from Slack support team that you can not deactivate file sharing in admin settings.
But I think I might have a solution for you. You can create a Slack bot that automatically blocks sharing of files in channels. (technically the bot would auto delete any message that is a file share form a user). Users would still be able to upload files, but they wont be able to share them. At least not in any public and known private channel. However there is a caveat: As I understand the bot would not be able to block users from sharing files in direct messaging channels though.
We are using a similar bot to block messages from unauthorized users in some private channels, which works nicely. Our bot is a simple Slack app with a bot user and using the Events API to receive all messages from all channels he is invited too. The bot then checks each new message and deletes the ones that does not fulfill some criteria. In our case if the message comes from a user that is not on the whitelist.