Downloading OSM data for multiple polygons at one time

Good day, I am interested in the number of buildings found within each protected area in Africa. I have a protected area layer (WGS84) that I am using. The amount of data per protected area will be very small, most likely less than 200 nodes per protected area. However, I cannot find a GIS tool (either Q or Arc) that will allow me to download OSM data for multiple polygons from a vector layer at once, despite the data being very small.

Any assistance here would be greatly appreciated.