If you add strict and warnings, you'll get the error:

Argument "Attendance" isn't numeric in array or hash lookup at script.
+pl line 36, <DATA> line 1.
Argument "Sales" isn't numeric in array or hash lookup at script.pl li
+ne 36, <DATA> line 1.
Argument "Tickets" isn't numeric in array or hash lookup at script.pl 
+line 36, <DATA> line 1.
Argument "Fans" isn't numeric in array or hash lookup at script.pl lin
+e 36, <DATA> line 1.
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When you do your hash access, $_ contains the entries of @report_name, not the indexes. Therefore $report_name[$_] is, for example, $report_name['Attendance'], which returns the 0 element, Attendance. So grab the value for Attendance repeatedly. You can fix this with

#!/usr/bin/env perl
use strict;
use warnings;
use Data::Dumper;

my @report_name = qw(Attendance Sales Tickets Fans);

while (<DATA>) {

        my %by_report = split ' ';
        print Dumper(\%by_report);
        for (@report_name) {
        print "report_name is: $_\n";
                if ( exists ($by_report{$_} ) ) {
                print "$_ Report is: $by_report{$_}\n";
                } else {
                print "No data for $_ Report\n";
                }
        }
}
__DATA__
Attendance 7424 Sales 7074 Tickets 350 Fans 109
Attendance 321541 Sales 321508 Tickets 31 Fans 101
Attendance 9208 Sales 9208 Fans 102
Attendance 9832 Sales 9829 Tickets 3 Fans 106
Attendance 1559058 Sales 1503174 Tickets 55404 Fans 104
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Even better would be naming your loop variables (see Foreach Loops), to avoid confusion. You should also see Use strict warnings and diagnostics or die to understand why strict is your friend.