Adobe Flash is required to upload files in many situations, and it is sometimes blocked by Chrome. There is often no obvious evidence of this other than the fact it isn't working.
Fixed it by following the instructions to "Turn On Adobe Flash Player" as per the Chrome instructions at https://support.google.com/chrome/answer/6258784
For best results, add the website URL as an exception to the list.
The below snip indicates a simple way to achieve this as of July 2017. Chrome's workflow is subject to change at any time without notice, so your experience may vary.