This should do it:

#!/usr/bin/perl
use warnings;
use strict;

use Fcntl ':seek';


open my $CSV, '<', 'donor_82_01.csv' or die "Cannot open 'donor_82_01.
+csv' because: $!";
open my $TAB, '<', 'tmp12' or die "Cannot open 'tmp12' because: $!";
open my $OUT, '>', 'tmp12_01' or die "Cannot open 'tmp12_01' because: 
+$!";

my $pos = tell $CSV;
my %csv_data;
while ( <$CSV> ) {
    my ( $first ) = split /,+/;
    print $OUT ",$_" unless length $first;
    push @{ $csv_data{ $first } }, $pos;
    $pos = tell $CSV;
    }

while ( <$TAB> ) {
    my ( $first, $second ) = split or next;
    next unless exists $csv_data{ $second };
    for my $pos ( @{ $csv_data{ $second } } ) {
        seek $CSV, $pos, SEEK_SET or die "Cannot seek on 'donor_82_01.
+csv' because: $!";
        print $OUT "$first,", scalar <$CSV>;
        }
    }

close $CSV;
close $TAB;
close $OUT;
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