Can Rek'Sai Void Burrower use Hall of Legends to ready and then exhaust to use its Conquer ability?

No, you cannot use Hall of Legends to ready Void Burrower so that it can then be exhausted to trigger its own "When you conquer" ability within the same sequence.

Explanation

The reason this does not work is due to how Conquer Effects and the Chain function:

1. Triggering: When you conquer Hall of Legends, both Void Burrower's "When you conquer" ability and Hall of Legends' "When you conquer" ability trigger simultaneously.
2. Chain Order: Because you control both triggers, you must declare the order in which they are placed on the chain at the start of the sequence (Rule 383.3.d).
3. Timing Constraints: Even if you order the Hall of Legends trigger to resolve first to ready Void Burrower, Void Burrower is already exhausted at that point because it was the source of the conquer (assuming it was part of the unit(s) that achieved the conquer).
4. No "Mid-Resolution" Exhaustion: You cannot "pause" the resolution of a chain item (like the Hall of Legends effect) to perform an action (like exhausting Void Burrower to trigger its ability) in the middle of that resolution. All instructions within a chain item must complete before the game checks for new triggers or allows new actions.
5. Trigger Timing: For Void Burrower's "When you conquer" ability to trigger, it must be ready (or available to be exhausted) at the moment the conquer event occurs. If it is already exhausted as part of the combat resolution that led to the conquer, it cannot be exhausted again as a cost for its own ability (Rule 414.1.b).

Summary

While you can use Hall of Legends to ready Void Burrower after a conquer, doing so happens after the event has already occurred. Void Burrower cannot be readied and then exhausted to pay for its own conquer trigger in the same instance, because the opportunity to trigger the ability happens at the moment of the conquer, at which point the unit is already designated as having participated in the combat/conquer and is exhausted.