I can't reproduce your problem.

My script, as posted above, does not do that, in fact, it never submits because that's commented out. To help me help you better, you could post a minimal version of your script that still reproduces the problem but is no longer than 20 lines. That would enable me, again, to reproduce your problem and maybe find a diagnosis that helps us determine the best approach to a solution.

My apologies. Here's the code example. It inputs the text into the text box and selects the correct radio button and then Firefox goes to 100% usage on a single core.

use strict;
use WWW::Mechanize::Firefox;

my $url   = "http://tools.immuneepitope.org/tools/bcell/iedb_input";
my $agent = WWW::Mechanize::Firefox->new(
    activate  => 1,
    autoclose => 0,
);

my $method = 49;
my $seq = "ARRRSFASDATRASDFSDARASDAGADFGASDRFREWFASCDSAGAREW";

$agent->get($url);
if ( $agent->success() ) {
    sleep(5);
    print "Retrieved $url\n";
    $agent->form_name('form1');
    #$agent->set_fields( 'sequence' => $seq,'method' => $method);
    $agent->field( 'sequence' => $seq );
    my $q = sprintf '//input[@name="method" and @value="%s"]', $method
+;
    my $method_field = $agent->xpath( $q, single => 1 );

    $agent->click( $method_field, synchronize => 0 );
    sleep(5); 
    $agent->submit();
}
[download]

Ah - again, it's my error, as the API for ->click() does not want a hash but a hash reference when passing in elements and instructions for "blindly" clicking things. I've taken out your calls to sleep and added some warnings for the progress. The main change is from

    $agent->click($method_field, synchronize => 0 );
[download]

to

    $agent->click({ dom => $method_field, synchronize => 0 });
[download]

use strict;
use WWW::Mechanize::Firefox;

my $url   = "http://tools.immuneepitope.org/tools/bcell/iedb_input";
my $agent = WWW::Mechanize::Firefox->new(
    activate  => 1,
    autoclose => 0,
);

my $method = 49;
my $seq = "ARRRSFASDATRASDFSDARASDAGADFGASDRFREWFASCDSAGAREW";

$agent->get($url);
if ( $agent->success() ) {
    #sleep(5);
    print "Retrieved $url\n";
    $agent->form_name('form1');
    #$agent->set_fields( 'sequence' => $seq,'method' => $method);
    $agent->field( 'sequence' => $seq );
    my $q = sprintf '//input[@name="method" and @value="%s"]', $method
+;
    my $method_field = $agent->xpath( $q, single => 1 );

    warn "Method";
    $agent->click({ dom => $method_field, synchronize => 0 });
    #sleep(5); 
    warn "Submitting";
    $agent->submit();
}
[download]