Tip #1 from the Basic debugging checklist: use warnings. You were trying to grab an array element by using a string value instead of a numeric value, which is what the warning message tells you.

use warnings;
use strict;
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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