Can I use Ahri Inquisitives ability more than once in the same turn if there are multiple showdowns?

Yes, you can use Ahri, Inquisitive’s "When I attack or defend" ability once per showdown.

According to Rule 383.4.f.2.a and Rule 383.4.e.2.a, "When I attack" and "When I defend" triggered abilities are limited to triggering once per combat. Because per FAQ #6149, you can initiate multiple showdowns at the same battlefield in the same turn, and each showdown constitutes a separate combat, the ability triggers once for each of those combat sessions.

In summary, Ahri, Inquisitive’s ability triggers the first time she gains the attacker or defender designation in each individual showdown.

Note: I could not fully verify this ruling against the retrieved rules.