Will our application server have issues connecting to a Postgres 11.4 server with a 9.6 client? Our application server is on EC2 running on Amazon Linux 1.  Postgres dropped support for Amazon Linux and so we depend on Amazon providing the postgres client.

The client is 9.6 and our Amazon hosted RDS postgres server is 11.4.  Because of our automation, it would take a good amount of time to upgrade from Amazon Linux 1 to Amazon Linux 2.  Even then Amazon Linux 2 only has a postgres 10 client.

It's a small app that uses the Rails ORM Active Record and only does simple queries and inserts.

Is it a no-go to use 9.6 as the client?  I'm wondering what the risk are.
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.