Question: Wondering who actually built the Starlink propulsion systems? 

Did SpaceX develop this technology in-house, outsource or acquire a company that had the IP and bring this back in-house?

Very curious as to who developed their tech.

Answer: 
SpaceX developed a Krypton Hall thruster completely in house. Their experts had experience from other places. I know quite a few of the experts came from the University of Michigan PEPL laboratory. 
<https://www.linkedin.com/in/rafael-martinez-89b0329b/>
The krypton engine used in starlink satellites could have been developed in Poland. at the Institute of Plasma Physics and Laser Mixing. It is important that the working factor is krypton, which is many times cheaper than xenon.
<http://www.elektroonline.pl/news/4555,Polacy-zbudowali-elektryczny-silnik-plazmowy-napedzany-kryptonem>