They are called "Blind Credits".
I believe most merchant accounts allow this type of transaction. Usually you must be approved and security is a high concern.
I am currently exploring using NPC.net and they do allow Blind Credits. To use them, you must collect all the same information you would need to charge the person, then a negative amount is used in the charge field.
Well you can start with PayPal, that is probably the easiest for "anywhere in the world" transfers from your account to any combination of their accounts. Their fees are at the high end though.
The last time I researched it, a few years ago, there are a wide range of other options depending on how much work your willing to put in, most are cheaper but require more effort and have more restrictions in terms of accounts and countries they could work with ... plus they tried to push the risk onto you more than PayPal.