Logging off/out of a given user account, stops all applications (i.e., Processes) the user has/had running. Period.
If the client now needs for the automation to run as a Service (not necessarily an actual Service, of course), then the client workstation (aka user account) needs to initiate the Process via an interface with said Service running on a Service host (another machine, presumably a server, or workstation dedicated to same, which has it's own implicit licensing requirements).
You'll need to automate/schedule the start of the monitor Process (Service) on the Service host, and reconfigure your app suite to 'watch' for incoming requests from one or more clients, and implicitly configure clients to have sufficient AD/Share/NTFS permissions to 'post' the request wherever the Service host is 'watching', if that makes sense.
Same goes for the return process, or completion of same - the Service host now also requires sufficient AD/Share/NTFS permissions to write the data to appropriate locations either locally on the Service host, on the network, and also to send the resultant output to the printer, etc.
To boot, you must ensure that the Service host 'always' has the appropriate mapped network drives, and printer connections - I'd recommend the latter be performed via Group Policy using Print Management (printmanagement.msc + gpmc.msc are your friends), specifically setting the Printer's Deployment with a Connection Type of 'Per Machine' so that the connection is made available when the machine starts/restarts prior to any user account being logged on.
HTH