First of, note that what you try to do might be illegal since a number of sites prohibit the use of scripts to interact with them. You'll have to check the end user agreement to make sure.
Technically, a number of things went wrong, see the code below for something that seems to work:
my $URL = "http://www.arabidopsis.org/cgi-bin/Blast/TAIRblast.pl";
#BLASTX:NT query, AA db ->blastx
#AGI Proteins (Protein) ->ATH1_pep
#query sequence ->QueryText
my $response = $browser->post( $URL,
[ 'Algorithm' => 'blastx', 'BlastTargetSet' => 'ATH1_pep',
'QueryText' => $sequence,
'ReplyTo' => 'firstname.lastname@example.org'
The URL for the request was wrong, the first parameter is 'Algorithm' and not 'Algorithem' and you need to supply an email address, apparently.
Hope this helps, -gjb-