Is there anyway to get user activity within a date range for a given SharePoint site collection?
Seems like the Management API get's for all SharePoint only. It needs to be from C#.
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You cannot directly target a particular site collection but what you can do it use is Audit.SharePoint content type filter while make REST API call.
Once you have data from above service you can use filter based on SharePoint Base API Schema with parameter Site, you might have to use Site Id as parameter to filter events specifics to a particular SharePoint site collectoin.