Obviously, the donor may make the payment.  The question is whether he can claim a charitable contribution deduction for the payment, and the answer is yes.  A donor may claim a charitable contribution deduction for a gift “to or for the use of” a charitable organization, and it is not unusual for a donor to make a payment directly to a vendor to pay for goods or services used by a charity.

You should provide an acknowledgment that he made the payment to the vendor on your behalf. (It is a payment and not an in-kind donation.) You should also state whether he received any goods or services in return, and, if so, the value of such goods or services.

He could also give a check to you and you could use the money to pay the vendor.  That is a more usual way to do it, but either method will work.