All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad
is His slave and Messenger.
The fact that the merchant is non-Muslim or that his account is with an interest-based bank does not affect the permissibility or validity of the transaction. What then remains to be examined is the nature of the transaction itself in order to determine its ruling.
From what appears obvious from your description in the question: the store acts as an agent on your behalf to purchase the internet service for you in return for a fee. If this is indeed the case, then the transaction is permissible. Paying the amount to the store after the purchase has been completed does not negate its permissibility. This is because, even if we were to consider the amount paid by the store from its own account as a loan to the customer for the price of the internet service, the combination of a loan and a fee without any stipulated condition is not prohibited.
If this does not accurately reflect the reality of the transaction, then it is requested that the question be resent with a clear and specific description of the transaction, free from ambiguity.
Allah knows best.