There are not physical ‘horizontal plates’ for the Allen Mouse Brain Reference Atlas. The planes of view are virtually rendered, based on data gathered from coronal images that are averaged to a non-physical reference volume. Refer to the Documentation or technical whitepapers for the Allen Mouse Brain Connectivity Atlas to learn more about the creation of the common coordinate reference atlas framework or CCF. They can be downloaded directly here.
There are some recent publications describing reference atlas utilities from scientific teams not affiliated with the Allen Institute that you may be interested in. A publication by Xiong et al gives some nice use case examples for mapping that can be done without 3D reconstruction. Another publication by Puchades et al. describes the creation of a tool for reference mapping.