Sometimes when a package is received, there is no packing slip. Is it acceptable to write on the invoice the following? “No packing slip” and “Received by” and the person’s name who received the package and the date received?

Yes. The goal is to have documentation that the items were received, so indicating on the invoice that the items were received, the name of the person who received them, and the date of receipt is acceptable in lieu of signing the packing slip.