Is the benefit derived from cashback on a Visa card, in the form of an additional amount upon purchase, considered unlawful (haram) or usury (riba), whether using my own money or using clients' cash to pay their dues via the card to obtain the cashback?

Using bank cards is permissible if they are free from usurious conditions. It is also permissible to benefit from cashback when paying with them. There is nothing to prevent this, even if you pay with them on behalf of someone else. It is preferable, however, that you stipulate to the person for whom you are paying the cost of purchases that the cashback should be for you.