Interesting question. I’m not aware of any rules specifying that a card number needs to be on a card, the type of transaction you’re referencing is called a ‘card present’ transaction. When a card is physically present and swiped/inserted into a machine to read off the numbers.
There is a separate class of transaction called ‘card not present’ , and those typically refer to online transactions when a customer keys in their credit card details into an app/browser for online shopping.
Since most customers don’t have card readers at home, the bank typically prints these on the card to enable customers to read their card details while shopping online.
There is however a CVV2 number, typically a 3-digit number printed on the back of the card. The CVV2 is never stored electronically on the mag stripe or even the chip, it is just printed on the card and meant to facilitate ‘card not present’ transactions. This number is what merchants typically request for online transactions.
If you’re worried about someone reading your card number and then using it to shop online, you can actually just scratch the CVV2 number off. That way, it’s worthless for an online transaction.
‘Card Present’ is much harder to carry out, but not impossible. However to answer your question, I doubt (very much) that any bank will issue you a card with no embossed number on it, no harm trying though.