I wanted to check if it’s possible for us to set up the manager software on both our computers in a way that both can enter data at the same time, and the data syncs between both systems. Or would we need to opt for the cloud version to achieve this?
You need either the Cloud or Server edition of Manager
As you do not have the technical expertise to install, administer and manage the Server edition (your question is evidence of this) then you need to use the Cloud edition