He puts a certain amount of money on his credit card and buys things with it through the Internet

Praise be to Allah.

Firstly: 

It is permissible to buy things through the Internet, apart from things in which it is stipulated that the transaction takes place hand to hand, such as gold and silver, in which case if their reaching the buyer will be delayed, it is not permissible to buy them through the Internet. The Islamic Fiqh Council has issued a statement that it is not allowed to buy gold using credit cards because the immediate hand to hand exchange does not take place in that case. See question number 89787. 

Secondly: 

There is nothing wrong with using a credit card on which the customer has put the money he wants, and there is nothing wrong with the bank taking the fee mentioned in return for offering this service. 

And Allaah knows best.