Learn Hardcore Greek: Αν χρειαστεί επιστροφή, θα μου ζητήσουν την απόδειξη στο κατάστημα. - If a return is needed, they will ask me for the receipt at the store.

Αν introduces a real/possible condition: if something happens or is needed.
Here it means “If a return is needed/required …” (a realistic possibility), not a hypothetical like “If I were rich…”.

What is χρειαστεί exactly, and why not χρειάζεται?

χρειαστεί is the aorist subjunctive form of χρειάζομαι (to need / to be necessary).
After Αν, Greek commonly uses the subjunctive to talk about a possible future situation:

Αν χρειαστεί … = If it becomes necessary / if it’s needed (at some point)

χρειάζεται would be present indicative and would tend to mean something like:

Αν χρειάζεται επιστροφή … = If a return is needed (as a general fact/right now)
The sentence is about a possible scenario, so χρειαστεί fits better.

Why is there no word like να before χρειαστεί?