When you say:

#!/usr/bin/perl
  `intranet/develop/Susp/mifuture.cgi idnumber=218`;
  `at now + 2 minute <<EOF`;
  `/....../mifuture.cgi idnumber=220`;
  `EOF`  ;
  $atq = `atq`;  print qq( ATQ: $atq);
[download]

each backticked line runs in sequence. Try this:

(NOTE: Untested but close, and right in principle.)

#!/usr/bin/perl
  `intranet/develop/Susp/mifuture.cgi idnumber=218`;
  `at now + 2 minute <<EOF
  /....../mifuture.cgi idnumber=220
  EOF`  ;
  $atq = `atq`;  print qq( ATQ: $atq);
[download]

or something like it. And read up on backticks and system().

UPDATE: My solution will work (I think), and you can learn from studying it, but in all honesty, I think this answer from Fastolfe is better--I'd somehow overlooked piping.