Doable, YES.
Recommended, IMHO NO.
1. The main point is that the server side trigger framework depends on "quick responding" triggers scripts, as a running script is a "blocking" single task.
The export of document can take up to minutes, based on the size of the document and it's attachments. During the export no other trigger script can be performed which might lead to nasty user experience in other areas.
2. you would need a installed client (that contains the gateway) on your server machine. Performing exports on your server would also store all the temporary data that is needed on your server machine. So you would need extra maintenance and care to cleanup your server's file-system after an export.
We use a design where the trigger in integrity only "fires" and event that is performed by a second server (Jenkins). The Jenkins Server performs the export just as any other build job in a temporary workspace and cares for archiving result and cleaning up temporary files itself.
HTH Matthias