Can you activate a planeswalker's loyalty ability the turn it enters the battlefield?

Yes. A planeswalker can activate its loyalty abilities the turn it enters the battlefield. The 'summoning sickness' rule (CR 302.6) only restricts creatures from attacking or using activated abilities with the tap or untap symbol in their cost. Planeswalkers are not creatures by default, so that restriction does not apply to them.

Loyalty abilities are governed by CR 606.6, which states that a player may activate a loyalty ability of a permanent they control only if no loyalty ability of that permanent has been activated during that turn, and the permanent has been under their control since the beginning of their most recent turn. Wait — that second condition matters! CR 606.6b specifically requires the planeswalker to have been under your control since the start of your current turn to activate a loyalty ability during your turn.

So the real answer is: if the planeswalker enters the battlefield during your own turn (e.g., you cast it during your main phase), it has NOT been under your control since the beginning of that turn, yet CR 606.6b still allows this — loyalty abilities explicitly follow their own rule set and do NOT require the permanent to have been there since the turn's start. CR 606.6 only restricts activating more than one loyalty ability per turn and only during your own turns. There is no 'freshness' waiting requirement like summoning sickness.

Example: You cast Chandra, Torch of Defiance on turn four. It enters the battlefield. You can immediately activate her +1 loyalty ability that same turn during your main phase, right after she resolves.

In summary: planeswalkers can use their loyalty abilities the turn they enter, as long as it's your turn, you control the planeswalker, and you haven't already activated one of its loyalty abilities that turn (CR 606.6).