Yes Possible. If an automata is not acceptor but transducer then final state is not needed.
Any class of an automata can be without a final state! An automata can be thought as a finite representation of a formal language(that can be infinite set). An automata with the final state(s) is called acceptor. For example, A DFA as acceptor either accepts or reject a string and represents a regular language.
But another model of automata is called transducer that may not have any final state. The purpose of automata as a transducer is to produce output string for a given input string. Example for finite state machine as transducer is Mealy and Moor machine.