In looking at the documentation for AGOL, I'm afraid it doesn't seem so (I'm looking here: Overview—Web AppBuilder for ArcGIS | ArcGIS). You have some tools at your disposal, like the SearchWidget with FeatureLayer as the source, and the GeoLookup widget, but none of them would appear to do exactly what you need. It doesn't seem like you could use the Geoprocessing widget, either.
The process itself is certainly doable - I do some similar things with the JSAPI and ArcGIS Server, but you'd need to code a few things. Basically, you'd geocode an address using the search widget, then, on a processing/selection event, you'd execute a map service query using the returned x/y point geometry - you could check to see if is contained by, or intersects, the city limits polygon feature, and if so, return a response to the user.
One thing to consider as well, if the city limits polygon extent is coincidental with zip codes, you could do some simple processing in ArcMap to obtain the "sub regions" list. Then, you could do a simple JS look-up, on the front-end, against the zips list when the user enters an address... just make sure you validate the input first so they can't execute a search without zip. Technically, you wouldn't even need the address at this point, but, it would only be valid if zips did not bleed in/out of the city limits. If it's not coincidental, you might be 80, 90% valid, etc., but it won't be 100% accurate. Neither is geocoding, though! Even if the confidence score is 100, it may not be rooftop and could theoretically place the point outside of the city limits, when, in fact, it is not. Just some things to consider.
