Can you activate your chosen champion unit's ability while it is on the chosen champion's slot?

No, you cannot activate a champion's ability while it is in the Chosen Champion slot.

According to Rule 373, activated abilities can "primarily be activated while on the Board." Furthermore, the FAQ clarifies that the Chosen Champion Zone is not part of the board, and spells or abilities cannot interact with units in that zone unless they explicitly state they can.

Because an activated ability requires the unit to be on the board to be activated, a unit in the Chosen Champion slot is ineligible to use its abilities until it has been played onto the board.