I have an EB application with a print map tool in it that has been failing regularly lately. There's a lot of weird conditions going on here so I'm going to bullet point all the scenarios:
For reference, we use the standard ArcGIS print service, and all the data in the map we're trying to print is either from our own ArcGIS Online hosted feature services, or a different organization's (our county government) server hosted parcel layer.
I'd check the developer toolbar (f12) and check for errors/info in the tabs. It seems likely it's issues accessing certain layers with different user permissions.
I'll check the developer toolbar, but they were able to use the print map function for over a year with no issue before this started happening. None of the permissions have changed. And in the situations described above, the 2nd and 3rd bullet point describes different users with the same permissions.
I have seen some strange behavior from the print services. in Exb recently too.
Sometimes it requires creating a 'custom' print service when using a custom layout.
If it's failing from different computers I'd be curious if the routing in the network is the same for them when calling the service, or if something in between is blocking them also. when checking the Dev Tools via F12 look for any failing Network requests also.