I have configured an Apache httpd website with SSL client side certificates so that only users who have installed the correct certificate in their web browsers can access the we
The SSL (TLS) protocol only allows the server to specify two constraints on the client certificate:
You can use "openssl s_client" to see which CAs your Apache server trusts for client certs. I do not know how to configure Apache to change that list (sorry), but I bet there is a way. So if you can limit the list to (say) your own organization's CA alone, then you will have done all you can to allow a Web browser to select the client cert automatically.
As Eugene said, whether the browser actually does so is up to the particular browser.