Not a QGIS solution but what may work for you is the Python Geocoder library. For reverse geocoding you have the option to choose from multiple providers though some of them may have limits on number of queries or may require an API key.
Since you have more than 1000 features to look up, OpenStreetMap provider should do the trick without any restrictions.
Following is an example of how to go about it (assuming
points is a list of features you want to reverse geocode with their latitude and longitude stored as
points[index].long) while storing the address in
for index in range(0,len(points)):
g = geocoder.osm( [ points[index].lat, points[index].long ], method='reverse')
points[index].address = g.address