Yes, that's definitely possible.  How to best do it depends a lot on your particular requirements and resources.  I think there are two main ways for you to do that:

1) If you have a server that is strong enough (and, importantly, actually your server and not just some web host), you can run Bitcoin Core on it and use the RPC interface to manage the wallet, check for incoming transactions and so on.  If you do this, be careful that you know what you are doing and are not keeping too many coins in the wallet on the server.

2) You can pregenerate a list of Bitcoin addresses from some wallet (where the private keys could even be totally offline) and then just have a simple script on the website hand out one address after the other from this list to customers.  You could use some API provider (blockchain explorer with programmatically accessible API) to check for incoming transactions.  This method is probably simpler for you, but you have to trust that API provider "somewhat".

In any case, note that this will let you accept Bitcoins directly - without converting them to fiat currency on the fly.