I was able to run your code with example data. One potential problem is the line:

    if ( $config[$m]{Vendor} =~ /$products[$n]{Name}/ ) {
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which won't work right if any vendor name is a substring of another. Another potential problem arises if the URL is not defined. Perhaps there are other problems with the input data.

Here's a version of your code with some example data and some debugging, which works for me:

#!/usr/bin/perl

use strict;
use warnings;
use diagnostics;

my @products = (
    { 'Prod' => 'pencil', 'Name' => "Staples",   'Vendor Item ID' => "
+1", },
    { 'Prod' => 'paper',  'Name' => "OfficeMax", 'Vendor Item ID' => "
+2", },
    { 'Prod' => 'eraser', 'Name' => "Walgreens", 'Vendor Item ID' => "
+3", },
); 
    
my @config = (
    { 'Vendor' => "Staples",   'URL' => "http://staples/item=*", },
    { 'Vendor' => "OfficeMax", 'URL' => "http://officemax/item=*", },
    { 'Vendor' => "Walgreens", 'URL' => "http://walgreens/item=*", },
); 
    
my ($n, $m, $url);
    
print_products("BEFORE");

$n=0;
for my $el (@products) {
    if ( exists $products[$n]{"Vendor Item ID"} ) {
        $m = 0;
        for my $el (@config) {
            if ( exists $config[$m]{Vendor} ) {
                if ( $config[$m]{Vendor} =~ /$products[$n]{Name}/ ) {
                    $url= $config[$m]{URL};
                    $url=~ s/\*/$products[$n]{"Vendor Item ID"}/g;
                    $products[$n]{TaggedID} = $url;
                }
            }
            $m++;
        }   
    }       
    $n++;   
}           
        
print_products("AFTER");

sub print_products
{
    my ($msg) = @_;
    print "$msg\n" if defined $msg;
    my $count = 0;
    foreach my $prod (@products) {
        $count++;
        print "$count:";
        foreach my $key (sort keys %$prod) {
            print " \"$key\"=\"".$$prod{$key}."\"";
        }
        print "\n";
    }       
}
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And, since you asked for better ways to do it, here is the main loop the way I would have written it:

foreach my $prod (@products) {
    next if !exists $$prod{'Vendor Item ID'};
        
    my $viid = $$prod{'Vendor Item ID'};
    my $name = $$prod{'Name'};
                    
    foreach my $conf (@config) {
        next if !exists $$conf{'Vendor'};
                
        if ( $$conf{'Vendor'} eq $name ) {
            next if !exists $$conf{'URL'};

            my $url = $$conf{'URL'};
            $url =~ s/\*/$viid/;
            $$prod{'TaggedID'} = $url;
            last;
        }
    }
}
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