You (or the Server admin) can change the owner of the workbook/data source. Only Owners can manually refresh a data source (or a packaged workbook).
Why don't you just set up a scheduled refresh?
The report has an extract schedule, however sometimes it is needed to run in other times before said refresh, thus, the user has to do the "refresh extracts" in order to get latest data. Thing is, 2 users might have to do that, the owner/publisher and another person who is not owner/publisher.
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One way is to give other users the Project Leader permission for the project where those workbooks are published. Project Leaders are able to run extract refresh schedule (and even make modification of your workbook). Pls read more Understanding Tableau Nested Project Permissions