CGI programs typically get a single HTTP request.
HTTP 1.1 supports persistent HTTP connections whereby multiple HTTP requests/responses are made w/o closing the connection.
Is there a way for a CGI program (or similar mechanism) to handle multiple HTTP requests/responses on the same connection?
I am using Apache httpd.
Keep-alives are one of the higher-level HTTP features that is wholly dealt with by the web server. They are out-of-scope for CGI applications themselves.
Accessing CGI scripts through Apache mod_cgi works with keep-alive for me. The browser re-uses the same TCP connection to fetch the page and then resources referred to by it, without the scripts in question having to do anything special.
If you mean you would like to have the same CGI process handle one request and then the next (instead of the process ending and a new one being spawned), then I'm afraid that's not possible. The web server will intercept keep-alives and make them look like single requests before your scripts can do anything about it. (If you want to do that to improve performance, consider a different gateway interface, such as FastCGI or language-specific options like WSGI.)
SCGI sounds exactly like what you want. It is similar to FastCGI but a simpler solution to implement (the S stands for Simple :)).