Is it possible for a diprotic acid to have Ka1?

Yes, it is possible.

Explanation:

A diprotic acid is an acid that can release two protons. That's why it is called diprotic.

Monoprotic → Release one proton, for example Formic acid HCOOH

Triprotic → Releases three protons, for example H₃PO₄

Polyprotic → Release many protons, for example EDTA

it is a weak acid.

In the first equilibrum, it release proton, and the second is released in the second equilibrium. So the first equilibrium will have a Ka1

H₂A  +  H₂O  ⇄  H₃O⁺  +  HA⁻                Ka₁

HA⁻  +  H₂O  ⇄  H₃O⁺  +  A⁻²               Ka₂

The HA⁻ will work as an amphoterous because, it can be a base or an acid, according to this:

HA⁻   +  H₂O  ⇄  H₃O⁺   + A⁻²      Ka₂

HA⁻   +  H₂O  ⇄  OH⁻   + H₂A       Kb₂

the millers tale is the answer.