Sure, just set up a cookie jar using HTTP::Cookies
my $cookie_jar = HTTP::Cookies->new( );
my $ua = LWP::UserAgent->new;
$ua->cookie_jar( $cookie_jar );
my $request = GET "http://www.doubleclick.net"; # don't worry, it does
+n't get saved :-)
my $response = $ua->request( $request );
$cookie_jar->extract_cookies( $response );
Subsequent requests using the same user agent will send those cookies to the server. If you don't want them to be sent, use a new agent or clear the jar with $cookie_jar->clear
When you say the server requires that the client accept a cookie, presumably you mean that the cookie must be kept around in order to retrieve another page correctly. Because when serving the first page, the server has no idea whether the user will keep the cookie (i.e. send it back later). If you mean it must remember it and send it back, you'll want to reuse the user agent (and cookie jar) as mentioned above. If you want to preserve the cookie jar for later use outside your script, take a look at the save and load methods in HTTP::Cookies.
For a longer discussion of this subject, check out perldoc lwpcook and poke through the Q&A section under the HTTP and FTP clients topic.