The United States Postal Service does not require a return address on regular parcel post mail. The procedure for sending packages without a return address is exactly the same as sending them with one, except that you leave off the return address.
Package the shipment in a box or a wrap so that the contents are secure. The post office will not take boxes that make noise when the package is shaken, they will also not ship hazardous chemicals.
Apply a label with or write out the receiver's name and shipping address. In order to deliver the package, the post office does require information on where the package is going.
Take the package to the post office. At the post office you will be able to pay for shipping and send off your package. Exclude the return address if you don't want it visible.