Responding to planeswalker with doubling season in play
You can activate the ability immediately, before any opponent has a chance to respond, because Doubling Season's ability is not a triggered ability.

Doubling Season's relevant ability says this:


  If an effect would put one or more counters on a permanent you control, it puts twice that many of those counters on that permanent instead.


This is a replacement effect, as indicated by the word "instead". It modifies events as they happen. In this particular situation, it modifies the event of the Planeswalker entering the battlefield by making it enter with twice as many loyalty counters as usual. The rest works the same as always: it enters the battlefield and then you have priority first, with nothing on the stack, so you can immediately activate one of it's abilities.