Hi Ashley,
I found the reason is because your print gp service is setup as Asynchronous
You can find in this parameter in your url
Execution Type: esriExecutionTypeAsynchronous
I am not really sure do you need asyn or not, but nevertheless, if you prefer to use asyn Execution, you need to add this line of code in print widget, based on the documentation: Print | API Reference | ArcGIS API for JavaScript (search "async" you will find it)
Therefore, in the print template sample, if you add your print service url you also need to add line 03:
app.printer = new Print({
"map": app.map,
"async": true,
"templates": templates,
Hope this can help.
Roll Tide!