I think this is the structure you are looking for:

while ( my @order_fields= $query->fetchrow_array() ) {
   my $ordernumber = $order_fields[0] or die "Error: Expected a non-ze
+ro order number" ;
   my $url = "link.com/$ordernumber";
   ...
}
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fetchrow_array returns an ARRAY, and you must extract the element from that, even for the case where there is a single element.