Well, they shouldn't. I figured maybe they were doing something weird with buttons & clicks, but apparently not :( I spent some time on this problem, and I have experienced the same thing you have. It doesn't matter how I use Mech to submit the "Pick Primers" - it always spits out that error. I managed to find an older version of thier code (I can't find the currently running CGI, they have only released old ones). Here is the line:

  if ($query->param('Pick Primers')) {
      process_input($query);
  } else {
      confess "Did not see the 'Pick Primers' query parameter"
  }
[download]

So it's nothing fishy, I think the encoding is wrong or something. Their form encoding says:

enctype="x-www-form-urlencoded"

I wonder if that has anything to do with it? I passed that as an HTTP header and it made no difference. I think the only solution here is to use a proxy to sniff what your browser is sending and then use the same proxy to sniff what Mech is sending and try to reconcile the two.

Sorry I couldn't have been of more help.