Is there any built-in mechanism to convert files to PDFs when uploading them to the media library, or does someone know of functionality that can accomplish same?
No, there is no built-in way to do this.
You can build a module that converts the files to PDF using iTextSharp, PhantomJS or Puppeteer.
Once upon a time @adamnaj wrote a script to export media items to PDF using Sitecore PowerShell Extensions. Maybe you can use that idea to import them back into the media library after PDF generation.