If I repeat Bellows Breath, does it target twice so I can't activate Ezreal Prodigal Explorer?

No — Repeat helps you activate Ezreal, it doesn't block you.

When you pay the Repeat cost for Bellows Breath, you make choices for both instances during the Make Relevant Choices step of playing the spell (Rule 820.2). That means you choose enemy units for the first volley and enemy units for the second volley — both sets of choices happen before the spell even goes on the chain. Since you've now chosen enemy units twice with the same spell, you satisfy Ezreal, Prodigal Explorer's "chosen enemy units and/or gear twice this turn" condition.

This is consistent with FAQ #9742 on Frigid Touch + Repeat and Ezreal: targeting with the initial effect and again with the repeated effect counts as two instances of choosing.

So you can cast Bellows Breath with Repeat targeting enemy units, and once the spell is on the chain, you can exhaust Ezreal (if you have priority in an Open State or after the chain resolves) to draw 1 — condition met.