Unless you have some serious programming chops, that could prove a bit difficult. I'm not saying you should ditch hostgator, but maybe just look into some of their higher services (VPS/Dedicated Server). As what you are trying to achieve really does need a dedicated server (or VPS).
I can't seem to visit the hostingdiary.com (blocked at work...), so I can't really see how the service works or is meant to work to give you an idea on how difficult/easy it would be to use for your purpose, but here is a brief outline of the complications you'd ensue:
- User uploads the file to your account at hostgator
- You need to send that file and any relevant details for encoding to a third party location, all the while keeping track of where that file should be located on your hostgator account (as you may receive 2-3 of these at the same time)
- You need to either give details to the third party to send the file back to, or you have to receive notification of completion or ask the third party if it is completed yet
- Once completed and the file is received, you have to place it where you want it to be stored
It can be done, but the amount of effort you are going to put into it, will make the VPS or dedicated server worth the money on its own. Just my 2 cents.