Rob Allen has a post to his site talking about using Composer with shared hosting, showing how to use this popular tool even if you're on a shared hosting environment and don't have direct SSH or shell access.
I've seen this sentiment a few times now, so this seems like a good time to point out that you do not need SSH access to your server in order to use Composer. In fact, I don't run Composer on a live server (regardless of whether it's using shared hosting) and it's not on my list of things to do in the near future.
What you do need is a process where you handle your Composer dependencies on your own computer where you have PHP running.
He gives two possible solutions to the problem: either commit your dependencies or create some kind of build script that can execute the Composer install for you on deploy. He gives details on both of these solutions including the process for installing the dependencies with an automated FTP script (run at deploy rather than committed).