Can U.S. shopping centers be exported as a shapefile and if so, how?

If your question is: "Is there a web site where I can click a few buttons and out comes a shapefile with all US shopping centers" then the answer is no. But if you're willing to invest more time then it is possible.

The closest you'll get to "a web site where..." is the "Overpass Turbo" service (example: http://overpass-turbo.eu/s/1zTh) but this requires that you do some research as to which OSM tags you want (malls? supermarkets?), also Overpass Turbo will only output GeoJSON (not shape files but converters exist), and might refuse to run on an area as large as the US - you might have to run it in batches.

A more robust method is downloading a raw OSM data file that contains data for the US (e.g. from download.geofabrik.de), then run the "osmium" command line program (requires Linux) to filter out the objects of interest (requires the same research about what exactly you want that I mentioned in #1) and then convert them to GeoJSON.

Alternatively you can import the OSM data into a PostGIS database with osm2pgsql (both PostGIS and osm2pgsql are available on Windows but running on Linux is easier), and then use the standard PostGIS tool "pgsql2shp" to create a shape file based on a SQL query formulated to return all shopping centers.