Hi there,
Unfortunately, you cannot book your ticket without a credit card.
You can buy a ticket directly from the ticket counter in Shinjuku Expressway Bus Terminal, but it is too risky not to make a reservation for a seat in advance.
You can make a reservation without credit card information if you book either by a phone call or from the Japanese page of the reservation site. But it is only written in Japanese, so it would be a bit difficult for you to book a ticket from there.
Here is a phone number and the URL, just in case.
Phone number: +81-3-5376-2222
URL: https://www.highwaybus.com/gp/index
Hope this can be your help.
Have a nice trip in Japan!
Japan Trip Search