I'm currently running a study where I need to query the number of points/lines/polygons that are contained within a project boundary - intersecting lines and polygons need to be counted as well. The features are linked to an agency server (operations allowed: Map (display), Query (Identify), Data (Find)). It would be extra helpful if I could get an output table that preserved the attributes, but I will walk away with just summing the points/lines/polygons contained within a polygon.
The service will need to be published out as either a feature service or WFS. This will allow you to export the data or perform spatial queries/selection on it. If the service is not one of the two noted above then it is probably just a WMS service which essentially is just images of the data.
I assume there is a MapService or Feature Service available to you, something like:
In that case, you can register a GIS Server in the ArcCatalog: As per above example enter only: http://services.arcgis.com/oKgs2tbjK6zwTdvi/arcgis/rest/services
and import data into ArcGIS Desktop. Then, you can work with the service or export the data to your local hard drive.