Depends on how the site is written to handle it. There’s plenty of shopping carts, for instance, that do this. Other stuff, too. Here’s a discussion of how it can be done with PHP on the server side. There’s other options, as well.

https://stackoverflow.com/questions/42379157/can-sessions-work-without-cookies-if-so-how-does-a-session-work-without-cookie

More info:

https://www.w3schools.com/php/php_sessions.asp