I'm using the Stripe API on a couple of projects.
For the first time I've to think about a special case where a customer registers his card (or whatever else if it's possible) and get some money from Stripe ; something like a refund but there's no payment before ...
I'm conscious about the risk of such a functionality for the system, it has to be overprotected from everywhere.
I'm searching the Internet and don't find any answer about it. Is it possible ? If yes how do you do that with the API ?
It is possible to issue Refunds via the Stripe API as outlined here:
https://stripe.com/docs/api#refunds
... but the business practice you are describing is disallowed in all Credit Card Processing contexts.
When using a service like Stripe or Square to accept payments, your payments flow through their Merchant Account, and within the context of Credit Card processing, a Merchant Account cannot be used with the intent to transfer money to random Cardholders as a business practice, except to refund money that was previously paid by a Cardholder as part of a prior Sale.
A merchant account is not like a bank account, but instead more like a line of credit. You could conceivably attempt to send money via refunds in spite of the rules, as nothing is barring you from trying; but eventually your transfers would get flagged, as the negative Merchant Account balance your business accrues would trip all sorts of warning flags throughout the Payment Processing chain. Eventually, you should expect to eventually be kicked off of Stripe if you didn't stop. Even if you were to enroll in your own through a Merchant Service Provider (Braintree, Auth.Net, etc.), having your own Merchant Account would not help. The rules are not very well documented for why this is disallowed, but the short reason is that there are fraud schemes that exist that are possible that can cost the upstream parties a lot of money in a very short period of time. It is not in their interest to allow this sort of refund practice. If you tried to setup your own Merchant Account, any knowledgeable Merchant Service Provider would reject your application if it became clear your intent was to primarily issue refunds to Credit Cards.
Instead of giving money straight-out to customers, if you are trying to offer some sort of enrollment or purchase bonus, it is better to offer a credit or coupon or other discount that doesn't change the semantics of only accepting payments from your Stripe / Merchant Account.
If you are trying to create some sort of money transfer service over Stripe, I strongly discourage you from using Credit Cards for receiving parties and recommend you look into performing ACH/Wire transfers instead. There is no corresponding restriction to money transfer in the ACH system as there is in Credit Card processing.