Data Refresh Concern with Refresh Date and Time

I have a number of reports that refresh multiple times daily through an enterprise gateway.  I recently added a "Last Refresh" table that uses the "local time" from a website.

I noticed that my datasets stopped refreshing and think it could be because my computer is offline during some of the scheduled refreshes for this "web call".

I am looking to understand why the refresh is failing, and how I may add a refresh date/time that will persist when my personal gateway is not online.

Thanks!