Background:
I am updating an internal application to a two-step authentication process. I want to add a client certificate authentication process
In every browser I've seen, the browser will not prompt you to select a certificate if it does not have any certificates signed by a CA the server trusts. So make sure your server is configured with the correct CAs. As Boklucius suggested, you can use openssl to examine the list of trusted CAs your server is sending to clients and see whether the CA you have signed your client certificates with is among them.