Even though it includes rolling a die, Vexing Puzzlebox's second ability is a mana ability. It does not use the stack and cannot be responded to.

If an effect instructs you to roll more than one die and ignore one or more of them, the result is only what wasn't ignored.

If an effect instructs you to roll one or more dice and add some number to that roll, the result is the total after adding that number.

In a game of Planechase, the result of rolling the planar die is not a number and does not cause Vexing Puzzlebox's first ability to put charge counters on it.