To get you started.

...feeding the data to the online tool...

Have a look at the LWP and WWW-Mechanize

...getting the results...

Have a look at HTML::TokeParser

Update:
Added WWW-Mechanize

Update 2:
This gets the html:

#!/usr/bin/perl

use strict;
use warnings;
use WWW::Mechanize;

# field name: inseq

my $url = q|http://thr.cit.nih.gov/molbio/hla_bind/index.shtml|;
my $seq = 'EALLKQSWEVLKQNIPGHSLCLFALIIEAAPESKYVFSFLKDSNEIPENNPKLKAHAAV
+IFKTICESATE
LRQKGQAVWDNNTLKRLGSIHLKNKITDPHFEVMKGALLGTIKEAVKENWSDEMCCAWTEAYNQLVATIK
AEMKE';

my $mech = WWW::Mechanize->new()
 or die "couldn't get Mech object: $!";
 
$mech->get($url)
  or die "couldn't 'get': $!";

$mech->submit_form(
  form_number => 1,
  fields      => {
    inseq => $seq,
  }
) or die "couldn't submit form";

my $html = $mech->content()
  or die "content failed: $!";

{
  my $file = 'bio_output.html';
  open my $fh, '>', $file
    or die "can't open $file: $!";
  print $fh $html;
  close $fh;
}
[download]

The error checking may be a <cough> tad excessive :-)

Update 3
Here's my go at extracting the data:

#!/usr/bin/perl

use strict;
use warnings;
use HTML::TableExtract;

my $html;
{
  local $/;
  my $file = 'bio_output.html';
  open my $fh, '<', $file
    or die "can't open $file: $!";
  $html = <$fh>;
  close $fh;
}

my $t = HTML::TableExtract->new();

$t->parse($html);

my $report = $t->tables_report(1);

print $report;
[download]

output:

---------- Capture Output ----------
> "C:\Perl\bin\perl.exe" monk18.pl
TABLE(0, 0):
User Parameters and Scoring Information:
method selected to limit number of results:explicit number  
number of results requested:20
HLA molecule type selected:A_0201
length selected for subsequences to be scored:9
echoing mode selected for input sequence:Y
echoing format:numbered lines
length of user's input peptide sequence:145
number of subsequence scores calculated:137
number of top-scoring subsequences reported back in scoring output tab
+le:20

TABLE(0, 1):
Scoring Results:::
Rank:Start Position:Subsequence Residue Listing:Score (Estimate of Hal
+f Time of Disassociation of a Molecule Containing This Subsequence)
1: 2:ALLKQSWEV:1930.068
2: 95:KITDPHFEV: 795.962
3: 108:LLGTIKEAV:  57.937
4: 107:ALLGTIKEA:  42.278
5: 21:CLFALIIEA:  42.278
6: 19:SLCLFALII:  16.254
7: 63:TICESATEL:  12.043
8: 12:KQNIPGHSL:   7.581
9: 14:NIPGHSLCL:   2.937
10: 101:FEVMKGALL:   1.911
11: 133:NQLVATIKA:   1.864
12: 35:YVFSFLKDS:   0.970
13: 45:EIPENNPKL:   0.903
14: 75:GQAVWDNNT:   0.756
15: 135:LVATIKAEM:   0.739
16: 103:VMKGALLGT:   0.737
17: 52:KLKAHAAVI:   0.524
18: 123:EMCCAWTEA:   0.457
19: 3:LLKQSWEVL:   0.434
20: 89:SIHLKNKIT:   0.420
[download]