Hi Andy,
I experimented with Experience Builder Beta and want to pass on what I found:
The Search widget is available once a Map widget is added. The Search widget is not configurable in the same way as the Web AppBuilder widget, and instead looks at the settings configured on the Web Map layer search. Fields with domains are not available to configure in Web Map search, either (known limitation).
I ultimately got this working in Experience Builder and the Web AppBuilder Search widget by calculating a new field in the Web Map from the field with domains to a field that does not have domains ($feature.DomainField in Arcade). This way the values are still constrained by the domain but auto-suggest functions as expected in search.
Hope that helps,
-Peter