New Glen is their orbital rocket but it is not ready yet. Hopefully it will be ready for some of the launches, but it must hurt owning a rocket company and having to use another companies to launch your satellites.
From the project web page
"Project Kuiper has secured 80 launches from Arianespace, Blue Origin, SpaceX, and United Launch Alliance, and we have options for additional launches with Blue Origin, providing enough capacity to deploy the majority of our satellite constellation. The agreements comprise the largest commercial procurement of launch capacity in history, and support thousands of suppliers and highly skilled jobs across the U.S. and Europe."
It's worth mentioning too that BO is following the same strategy as SpaceX (and others), in becoming their own customers. Lesson learned from the 90's launch vehicle boom. Companies couldn't scale without customers, it's a vicious feedback loop. No customers because prices are too high, can't bring prices down without more customers. Either you have to become your own customer or you have to bootstrap. It's a long way to bootstrap and an expensive industry.
Though using ULA is kinda a bridge to the looking deadline[0]. So if they can't get satellites up now they won't have this means for being their own customer in the future.
From the project web page
"Project Kuiper has secured 80 launches from Arianespace, Blue Origin, SpaceX, and United Launch Alliance, and we have options for additional launches with Blue Origin, providing enough capacity to deploy the majority of our satellite constellation. The agreements comprise the largest commercial procurement of launch capacity in history, and support thousands of suppliers and highly skilled jobs across the U.S. and Europe."