Japanese banking and card business are quite different from, let's say those in USA and other countries, so let me give it a try to explain....
- "Visa Debit" is not a common card type offered by card issuing companies in Japan. For online shopping I think that what you would need to get is a Visa Credit card. (As "debit" product, there is "J-Debit" in Japan, which comes with the ATM card of many Japanese banks, but as those require PIN code entry at shops, I don't know if those are accepted as payment at online shops.)
- In Japan, credit cards function almost like "deferred debit" cards, in the sense that you use it to shop, then the amount gets (one or two months later) automatically deducted from your bank account that you register with. Of course you can revolve too, but it is NOT that you receive the credit card bills sent to you at the end of the month and pay with a personal check (which is the case in US, for example). So consider Japanese credit cards as ALMOST like a debit card.
- Another difference about Japan is that traditionally it has not been banks who issue cards, but separate card-issuing companies, who may be bank's subsidiaries (this is due to regulation that was lifted a few years ago, which banned banks from doing credit card business). Even though by now this regulation has been lifted and some banks are doing card business, it is not YET very common.
So of course, you can go ask at a bank if they can issue credit card with the account, but my experience so far is that banks, when you open an
account with them and start with a somewhat low balance, unless they see your monthly salary amount and other papers, are not very happy to issue credit cards. (Others please correct me if i'm wrong on this.)
Instead, you can go in two steps (1) getting a bank account, then (2) getting a credit card for online shopping.
(1) To open an bank account, you can go to any bank, as it suits you; meaning, people have been writing in on this forum about different features
of different banks (search with "bank" on this forum), like convenient locations or low fee for remittance, etc., so you can choose according to that, and also depending on the place you live, pick a bank that is convenient.
(2) And to get a credit card, you can go to places like Credit Saison (at Seibu department stores) or Tobu Credit (Tobu department store), even Japan Railway (they offer SUICA credit card), etc. a whole bunch of card issuing companies that are NOT banks. If there is a department store nearby where you live and where you shop often, then department store cards WITH Visa credit function might come in handy.