Have a look at the YouTube Analytics API:
https://developers.google.com/youtube/analytics/
There is a metric for retrieving the share count (shares
). Use dimensions (dimensions=video&metric=shares
). Alternatively, use filters if you want to retrieve the share count for a specific video: video==Zhawgd0REhA
. For more info, see:
https://developers.google.com/youtube/analytics/v1/available_reports