There is a pretty extensive API on the printer, but this would require the remote server to poll the printers. You could also make a small service that runs on the printer itself that sends the data to the cloud. All systems on the machine are connected through DBUS, but those API's aren't stable just yet, so the second option will probably require more work / upkeep.
There's a way, but it's not 'problematically'...
(..at least not, if the printer is connected to a network).
The possibilities are explained in this topic:
Inside the Ultimaker 3 - Day 2 - Remote access (part 1)
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