I've found the raw LORRI images taken by New Horizons at Ultima Thule but haven't been able to find raw MVIC images (or data from the other instruments). Does anyone know if these are currently publicly available in unprocessed form (or where they will be in future)?
All the New Horizons data is archived and available at the Small Bodies Node (SBN) of NASA's Planetary Data System (PDS) at this landing page. PDS is the main official repository for all NASA planetary data from either space-based or ground-based missions. Since PDS is used as the final archive, there tends to be a small number of data deliveries of large chunks of data. Looking at Table 3 of the New Horizons Data Management Plan (which covers the Pluto encounter and not the Extended Mission to Ultima Thule), data were typically delivered to PDS about 9 months after receipt on the ground. The pluto.jhuapl.edu site will likely be the best outlet for more immediate data releases if they are made available.