Can I say 'to acknowledge a payment?'

The business language to be used is:

Please acknowledge receipt of payment.

It is short and sweet, and to the point.

You do not have to say "when you receive payment" or any sort of auxiliary instruction. The phrase is understood by anyone who has run a business professionally, and that a "receipt" is to be issued to the payer.

It is like writing "RSVP", where you do not need to tell the person invited "please ...blah ... when you receive this invitation." Writing "RSVP" simply says, "Please respond" (in French).

Saying "... when you receive ..." is redundant.

However, I notice that when "professional businesses" write to mom & pop stores, they frequently write (in a style similar to),

Please issue receipt to {business name and address} upon receiving this payment.

Alternatively, they go a step further

Enclosed with this payment is a form to indicate receipt of our payment. Please acknowledge the payment by signing the form and mailing it back to us in the stamped and self-addressed envelope provided.

BTW, "self-addressed envelope"? Can an envelope address itself? Seems illogical to me.

Yes, but I would say "when you have received", not "when receiving"