What happens when the same application from different server tries to connect to the always on primary server In our environment we have an always on availability group and the reporting server is acting as a secondary replica with Synchronous Mode, manual failover. XSLT report editor that is a vendor application will connect to the application server which in turn connects to the database server. The application server (separate server)  connects to the listener to serve client connections. My question is if I set up a separate application server just for the XSLT report which in turn connects to the reporting, now will there be any problems if the reporting server for some disaster recovery becomes the primary where the Client application and the XSLT application will both  connect, both the application is same  that is installed on separate server one for client another for the xslt report.

I am not comfortable to set up transactional replication which is an additional overhead. Any advice will be greatly appreciated.
From a technical standpoint, there are no problems with multiple applications, running on multiple different machines, connecting to the same SQL Server instance.

If you experience the failover scenario that you described, it's possible that the reporting workload will interfere with your other workloads from a performance standpoint.  

In other words, all the long read queries your reporting app does could block queries coming from your other apps.  And in general the reporting app queries will require CPU time and I/O time that is normally solely used by the other app.

It's hard to say without more detail whether this would just slow things down a little, or bring your whole app grinding to a halt due to blocking or resource contention.