Safari can only upload photos to websites. It's a basic limitation. Most third party web browsers have the same limitation. I was kind of hoping this would change with iOS 8, but it hasn't. At least not yet.
The one exception (that I know of) is iCab Mobile. You have to download or use Open In to get a copy of your file into iCab Mobile first, but it's possible to upload the file after that.
It does not work with every site. There are lots of different ways to implement a file uploader on the web, and not all of them work with iCab. You also have to dig into iCab's network settings and make sure the uploader is set for iCab Mobile (not iOS). Inside the app go to Settings (the gear icon) > Network > Uploads via.
There are a few other possible workarounds, but they are painful. The best option is to see if the site/company offers a more modern way to send your resume. Say, a DropBox link or share URL to some other cloud service. Or even, gasp, email.
Good luck.