That's 'drop shipping'.
Drop shipping is a supply chain management technique in which the retailer does not keep goods in stock, but instead transfers customer orders and shipment details to either the manufacturer or a wholesaler, who then ships the goods directly to the customer.
If you don't label the goods with your brand then it's 'white label drop shipping', 'blind drop shipping' (lacking a return address).
If you label it with the client's branding then it's 'private label drop shipping'.
The third party is a retailer (usually an online one) - you are a drop-shipping wholesaler.
Example: Ingram is a very large book wholesaler in the USA. Individual book retailers can, from their website or their physical store, have Ingram ship books direct to their customer under their branding:
CONSUMER DIRECT FULFILLMENT
Consumer Direct Fulfillment service offers speed, efficiency, and cost savings for online retailers. Orders from your website are processed in one of our four state-of-the-art distribution centers and shipped to your customer’s doors—as if they came directly from you. ...
DIRECT TO HOME
With Ingram’s Direct to Home service, you can ship books directly from Ingram to most addresses in the world, including military APO and FPO addresses. Any book in Ingram's inventory can be delivered straight to your customers' doorsteps as if it were shipped from your store. ...
Ingram: Retailer Fulfillment [sic] Services