I think you want:
items.GroupBy(item => item.Order.Customer)
.Select(group => new { Customer = group.Key, Items = group.ToList() })
.ToList()
If you want to continue use the overload of GroupBy
you are currently using, you can do:
items.GroupBy(item => item.Order.Customer,
(key, group) => new { Customer = key, Items = group.ToList() })
.ToList()
...but I personally find that less clear.