If I use Thrill of the Hunt on Akshan on my opponent's Aurora trigger and steal their Aurora, will they still get to summon a unit?

Yes, the opponent will still get to summon a unit.

According to RiftJudge FAQ #306, once a triggered ability is on the chain, removing or stealing the source of the ability (the Dazzling Aurora) does not prevent that ability from resolving.

Sequence of Events:

1. Trigger: The end-of-turn trigger for Dazzling Aurora is placed on the chain.
2. Reaction: You play Thrill of the Hunt in response to the trigger. You banish Akshan, then play him to a battlefield (assuming you have a valid target/location).
3. Stealing: Through the effect of your actions (or if the "stealing" refers to a specific interaction like Here to Help as mentioned in the FAQ), the Dazzling Aurora is moved to your control.
4. Resolution: Because the ability is already on the chain, it resolves normally when its turn comes, regardless of the fact that you now control the Aurora.

The trigger does not check the status or controller of the source gear at the time of resolution; it only checks the trigger condition at the time it was initially placed on the chain.