Nope, not possible.

You use to be able to book AQ and WP (Island Air) inter-island code shares. You could pick AQ and WP as your seconed and third choices to search on. Occasionally (in the pre-go! fare war days) such segments would even be cheaper than booking directly with AQ/WP.

But now on the "improved" website you can hunt for UA and include Star Alliance flights. But, of course, since AQ and WP aren't Star Alliance members (instead having their own odd partnership with UA) their flights don't come up (except as codeshares when doing, e.g., from/to-the-mainland flights. For example, a popular itin that comes up when doing a mid-con to KOA is XXX-ORD-OGG (on UA) and then OGG-KOA on WP.

But for strictly interisland flights, no go

Although for comedy purposes it is sometimes amusing to see what routing (and at what price!) you're offered for a simply HNL-OGG for example.

(A reminder: UA is the last legacy carrier that still offers a roundtrip inter-island partner award ticket for 5K miles.)