When Ajani, Mentor of heroes is on the battlefield with 9 loyalty counters and Ajani's pridemate is on the battlefield, if the counters are removed to activate the planeswalker ability: You Gain 100 Life, does Ajani's pridemate gain 100 +1/+1 counters?

Negative. Ajani's Pridemate is a triggered ability that happens once per instance of gaining life, regardless of how much life is gained. So, if you gave Ajani's Pridemade lifelink, it will not give itself two counters when it hits something as you technically only gained life once - you just gained two life from that trigger.

So again, the simple response is, no, it would only gain one counter, not 100.

Sunbond is what your looking for here. It will give you 100 counters if you ultimate Ajani.