Does the right of the Bank to take a fee for an online transaction with VISA card?

the Bank is emitiram card, look at the conditions in your contract. Visa Commission does not take.

Exchange Bank is a bullshit :)

First, the General scheme of payments using cards somewhat trickier than described in the previous comments.
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The ultimate payer of a fee for payment by card (without a difference — after that is done or offline) is a merchant shop where you pay for the purchase. This Commission being deducted from the merchant acquirer Bank, i.e. the Bank that provides the merchant services accept payments cards (the average is about 2%). Then the acquirer keeps part of this fee received, and the rest gives the network the payment system (e.g. Visa or MasterCard). Network payment system, in turn, leaves a portion of the money received and gives the rest to the Bank that issued the card — issuing Bank.
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As far as I know the current practice, a large part of the sawn of the Commission falls on the issuing Bank. And have never met and heard about a situation where the Bank takes the Commission for the implementation of the payment card (again, online or offline).

Overall, I agree with the view that feature in the contract — if the Commission prescribed, you can take.
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In General, curious to know — which Bank is very rare and, therefore, you can try to dig up the details.
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So in conclusion, it is necessary to distinguish transactions: if you make a transfer with your card (and not payment in the payment of a product or service) — the Commission for such operations is absolutely standard practice — then just need to choose price.