Debit cards can typically only be accepted in-person by using a local payment provider that provides hardware that supports debit card transactions. Outside of this, debit cards are not a support payment method online unless the debit card is registered to a credit card network. Visa/Mastercard debit cards work like normal credit cards but would withdraw the funds from a debit account instead of a line of credit. If the payment processor you use accepts Visa and/or Mastercard, then it should support purchases from customers who are using Visa/Mastercard debit cards so long as they have this feature enabled on their account.
You can view a full list of available payment providers for the United Ara Emirates here.