The only Puppy printer drivers that do allow authentication are for Windows shared printers. So here is something to try.
1. What Puppy version are you using?
2. Does it have PeasyPort in the network menu? If not, get it here.
3. Scan your network using the Names button. Does it detect your printer by its NETBIOS name?
Please explain. What Toshiba?In Toshiba, we can set Department ID and password in settings on PC