1. Perl is indeed wonderfully capable... but do you really need to do it the hard way? After all, a standard Perl remark is 'don't reinvent the wheel:' http://office.microsoft.com/en-us/templates/avery-5160-mailing-labels-featuring-avery-app-for-word-2013-anniversary-design-30-per-page-TC103749911.aspx
2. But if you're still going to do it in Perl, it looks as though you have a comma-delimited source file, AKA, CSV. So use one of the csv modules to drag in your data (not that what you were doing is terribly wrong).
3. And if you're still going to do it your way -- minus the formatting:

#!/usr/bin/perl
use strict;
use warning;      #mandatory inclusions until you know enough to know 
+when you can skip these helpers
use 5.016;

# Program name: perlReadAndFormat.pl
# Purpose: Open disk file. Read and display the records in
#          the file. Count the number of records in the file.


# open (FILEIN, "cust.txt") || warn "Could not open source file\n";
open (LABEL, ">","labels-to-print.txt") || warn "Can't create labels f
+ile\n";
# while (<FILEIN>)  
while (<DATA>){ 
     #print "$_"; 
     ($CUSTID,$fname,$lname,$phone,$address,$city,$state,$zip,$email) 
+= split(/,/,$_);
     # Or use array:  @fields = split(/,/,$_);
#    write(LABEL);  # send to output
     print LABEL "$CUSTID,$fname,$lname\n$address,$city,$state,$zip\n$
+phone,$email\n";
     ++$line_count;
}


print ("File \"passwd\" has $line_count lines. \n");
close (FILEIN);
close (LABEL);

__DATA__
$CUSTID,$fname,$lname,$phone,$address,$city,$state,$zip,$email
1,fred,jones,555-1212,123 Main St.,Boston, MA,00001,test@test.com
2,mary,smith,222-1515,321 Least St.,Hanover,NH,02022,testmary@test.com
3,jack,least,122-5511,231 Last St.,Franklin,CT,06001,nomail@test.com
[download]

Output:

1,fred,jones
123 Main St.,Boston, MA,00001
555-1212,test@test.com

2,mary,smith
321 Least St.,Hanover,NH,02022
222-1515,testmary@test.com

3,jack,least
231 Last St.,Franklin,CT,06001
122-5511,nomail@test.com
[download]

Obviously, this ignores your desire to spread them over the page; that's left as an exercise...
BUT IT WOULD HAVE BEEN NICE IF YOU'D SHOWN US THE DESIRED OUTPUT!