Can you take the shortcut on Koopa Troopa Beach without items?
Yes, you can take the shortcut it in every mode without an item. Doing a Mini-Turbo before going onto the ramp helps to increase your speed. Also jumping at the peak of the ramp can help on the slower speeds.
Here's a 50cc 1p GP mode example
Also you actually don't need the ramp at all to get into the tunnel.
Here's an example of that in 150cc 4p VS mode while small from a lightning hit (top right screen [player 2])
Yes, it is possible. As shown here.