We don't have a dedicated server. Is there a way to run VIP Task Manager from a shared computer?
Yes, sure. Any computer in the network can serve as a server and store the database. While installing the program, you can install "full installation" (in that case you will be able to work as user on this computer) or "server part" (in that case you will not be able to work as user on this computer) on one of your shared computers and use it as a server.
