What language is your site built in?

If you are using node, it won't be all that hard to write a layer on top of web3, and use Infura to broadcast it.

You'll need some ETH for the forwarding fees to move your tokens around, but apart from that, it is fairly straightforward.

If you are using PHP, it might be considerably more difficult to implement.

If you set it up right, you could add new deposit tokens just by adding their name, and contract address to a database.