There is nothing in settings that will change this. Safari only allows the uploading of photos or videos from the Photos app.
For the most part, apps are not allowed to access files stored in other apps. Photos is one of the few exceptions. It makes for a more secure device, but can also run into problems when it runs head first into some of the traditional way of sharing files on the internet.
If this is the only method the site provides to send them files (no email, links to DropBox, etc.) then there is only one solution I know of that that will (probably) work.
iCab Mobile. iCab is a third party browser that allows uploads to most sites. Once you install iCab Mobile, you have to copy the file to that app, most likely by using Open In from the app that currently has the file. Once a copy of the file is stored in iCab it will be available when you visit a website (in iCab) and select the Choose File button.
Because there are many different methods of providing file upload ability to a site iCab Mobile is not guaranteed to work, though they've dont a good job of supporting the majority.
