You need to find a card processor/acquirer that has implemented the relevant products/services from Visa and Mastercard and work with them to set up a card-to-card programme. You can either whitelabel the acquirer's license or get your own license.
There are different ways to do card-to-card transactions, from the older solutions like OCT / CFT (Original Credit Transaction / Credit Funds Transfer) to the newer Mastercard Moneysend and Visa Direct. There are pros and cons to both solutions.
IIRC, these are some that are able to offer some or all the services and can do custom card programmes: