Does our application work with sql-server AlwaysOn?
First, let's clarify terms. AlwaysOn is a Microsoft marketing term which includes both SQL Server Availability Groups and SQL Server Failover Cluster Instances. You should stop using AlwaysOn and be specific for all future communications.

If you are referring to FCIs: the configuration should be completely transparent to your application. They should configure the connection to the virtual network name of the FCI and your application will not notice any difference.

If you are referring to AGs: With AGs, they have the option of configuring an AG Listener or not. 

If they are not using a listener then all is well and your application won't notice any difference. 

If they are using a listener, in a single subnet, then it is transparent. 

If they are using a listener, with multiple subnets (multiple IPs for the listener), then your application will need to support the MultiSubnetFailover=TRUE connection parameter or they will need to modify the RegisterAllProvidersIP Windows property. If one of those aren't used then they will have issues with DNS caching the wrong IP address and might blame the application for failing to connect.

See this article regarding the MultiSubnetFailover parameter.

See this article regarding the RegisterAllProvidersIP property.