Will our application server have issues connecting to a Postgres 11.4 server with a 9.6 client?
I don't believe that PostgreSQL dropped support for any Linux distribution. Perhaps you mean that there are no binary installation packages provided.

Using a 9.6 client is no problem, since 9.6 is a supported release. You will not be able to use new features like scram-sha-256 authentication, but I guess you can live without that.

Building PostgreSQL from source would be another option.