On an average day, there is absolutely no need to book for travel between Tokyo and Osaka, you can just rock up, buy a ticket and get on board. The vending machines can be a bit complicated, so I recommend buying a ticket in person from the ticket office, there are no fees for this.
You will, however, want to book ahead during major holiday periods like New Year’s and Golden Week. The trains are also busier on Friday afternoons from Osaka to Tokyo, and Sunday afternoons in the opposite direction.
Also, while Japanese trains have historically had fixed pricing, they have become more and more dynamic in the last decade and there are now some pretty good discounts to be had if you book ahead. However, this usually requires committing to a specific train, so if you don’t know when you’re leaving, this is not going to be a great option.
Finally, there is no one Japan Rail website because JR is actually a loose consortium of a number of notionally independent regional companies. JR Central is the one that operates the Tokaido Shinkansen between Tokyo and Osaka: https://global.jr-central.co.jp/en/info/timetable/
Credit:stackoverflow.com‘