Crypto wallet for payments with no deposit We are wondering how we can build a wallet that talks to our website and gives the user the ability to request a withdrawal of his funds. We do not know what is the right question for this type of functionality that we want to build is, so any advice would be much appreciated: 

The flow would be the following -> 
a: User got rewarded in crypto (BTC, XRP, ETH, LINK) (a digit on the database). 
b: User withdraws his reward to their address.

The reward gets credited to the user's account (simply a digit on the database), and NOT to user's wallet address, as the wallet should NOT generate new addresses for deposits, as we do not want to allow deposits at all.

To help with the explanation, Bitcoin faucet websites have such functionality usually, where you can withdraw your earnings but not deposit funds in them.

Cheers
There are many layers to this question. Also, I'm going to assume using just Bitcoin, for simplicity. For other cryptocurrencies the procedure would be mostly similar with important differences mostly in the backend.

On the Bitcoin layer, the simplest solution would be to have a separate wallet for the website.

On the interface layer, you can have a simple form on your website where the user can input the address when requesting fund withdrawal.

On the backend layer, your website can use any wallet software with an API or CLI, Electrum should be good, but is just an example. This API or CLI could be then invoked to make payments to users.

Your server would necessarily have access to the private keys for the website wallet, therefore security should be an important consideration. The wallet file on the server needs to be protected and the backend of your website must properly check the withdrawal amounts, properly authenticate users and properly mitigate common attacks. If you don't have experience with website security, there are many potential holes that can open avenues for attack. Elaborating on them is beyond the scope of Bitcoin StackExchange. The important precaution would be not to keep more funds than necessary on the website wallet and deposit more as needed, manually, from a wallet on your own computer.

(This answer was edited by author as a result of a discussion in the comments)