We may support it in the future. Search implies selections, which the map viewer doesn't currently support. Then, you'll probably want to see the results, which means some sort of tabular display...
Hosted tiled map services currently don't support queries because they are just composed of a set of tiled images. Hosted feature services do support queries. ArcGIS Server map and feature services both support queries, unless the map service is just tiles, which is rare.
Thanks,
Mike