Suggestions:

Provide more information with $! when open fails (you'll apreciate it when it happens).

open (LOGFILE, "< $logfile") or die ("Could not open $logfile: $!");
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You're kind of care free when mixing $thiscust with record fields, hash keys and file handles. Maybe something more like this:

$thiscust = (split /\//, $fields[4])[1];
if ( ! $cust{$thiscust} ) {
    print "customer $thiscust\n";
    $cust{$thiscust} = $thiscust;
    open (my $fh, ">> ${prefix}${thiscust}${suffix}")
        or die ("Cannot open ${prefix}${thiscust}${suffix}: $!");
    $cust{$thiscust} = $fh;
}
my $fh = $cust{$thiscust};
print $fh $_;
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