The Sencha Open Source FAQ covers this use case.
There are 2 important points:
First, both the front end and backend source are likely derived works for the GPL as in this example from the FAQ:
Second, since ExtJS is run on the client they consider using the web app Conveyance:
Conveyance vs. Propagation
Since Ext JS, Sencha GXT and Sencha Touch are software programs that can run within the browser while disconnected from the network or the rest of a server program; when a Sencha based interface is embedded in a web-page served to a user who does not have an employee relationship with the original licensed entity, we consider that “conveyance” rather than simple web page “propagation” as defined by the GPL v3 has occurred, and the source code of the whole application must be provided to the user.
So you will need to make the source available to the users of the application. (Much like the AGPL.)
It is clear to me from reading their FAQ that if you intend to use ExtJS in a commercial app they want you to buy a commercial license.