Yes, this should be possible. The main part of the Leap Motion software is already a daemon. The only part with a GUI is the settings control panel. (After installing the software, you might have to disable the control panel from launching on computer startup to prevent windowing errors.)
For sending the data, you can use the WebSocket server provided by the LM daemon. This serializes the data as JSON messages. You do have to manually edit the daemon config file to allow remote connections to this server (see https://community.leapmotion.com/t/read-leap-data-from-guest-vm-on-host-machine-using-websockets/1523). This method is good if you are using JavaScript on the client computers. Otherwise, you can't take advantage of the APIs for other programming languages since they aren't set up to consume data as JSON messages.
An alternative to the WebSocket server is to setup your own data proxy server and send serialized frames (see Frame.serialize). You can then deserialize them on the client side and use any of the supported programming languages (except JavaScript).