When making a site that depends on a wallet, most people use bitcoind. This Daemon is part of the standard Bitcoin wallet, which means that it is very stable and secure. Fully making your own wallet from scratch might be possible, but would be very unsafe (and a lot of work).
bitcoind is a daemon that can be executed on Windows and Linux computers. When it runs, you can use RPC (Remote Procedure Calls) to create new addresses, send/recieve funds, check your balance etc. There's a lot of
documentation available on the Wiki.
Anyway, as a site you have two options:
1. Use bitciond, which does mean that you need a dedicated server that lets you run the daemon in the background.
2. Use an external service like the Blockchain.info API, which does this for you, meaning that you don't need a dedicated server. The drawback of this is that you are dependent on a third party, and when their site has any issues, yours will as well.
I imagine that the clients edited to work as altcoin-clients, have a very similar daemon program.
Hope this helps,
~W-M