Is it permissible, in Islamic law, for a service (charitable) office to take a percentage from each sale when sending clients to service providers, or to sell a card to clients to benefit from discounts, or to combine both methods? And are there other legitimate methods?

It is permissible to establish an office to offer discounts to customers in exchange for a percentage of the selling price. This falls under the category of mediation and brokerage. The conditions for this are that the fee must be known and the goods must be permissible. However, non-free discount cards are not permissible due to the elements of gharar (uncertainty) and maysir (gambling) involved.