How can I use openssl to get results from HTTP GET requests?

What you really should be doing is using a tool designed for fetching Web resources, such as curl, wget, or libwww-perl's GET command. If nothing is available, you should have your system administrator install something appropriate.

With that out of the way...

The openssl command does not terminate because the web server didn't close the connection.

Remember that by default HTTP keeps connections open after each request as a performance optimization. Once one request finishes, another request can be sent over the same connection, rather than closing and reopening a new connection.

If you want to instruct the server to close the connection instead, you can send the Connection: close HTTP header.