Is it possible to extract from my Pressero orders the information a user enters when customizing an eDoc template and pull that information into a file?

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It is possible to do this.  It would require both the Pressero and the eDoc API's. The Pressero API would be needed to get the session ID for the eDoc session and then the eDoc API would be used to extract the data from the session.  The program you write would need to place the extracted data into a file.