This seems like a relevant paper from 2017: Development of a 50,000s, Lithium-fueled, Gridded Ion Thruster.
Diagram shows a laboratory setup, so not something that will actually fly. I found no examples of those. Details are not given for how an actually deployed system might differ. It looks pretty much like a lab setup described in 2001 paper, Lithium Mass Flow Control for High Power Lorentz Force Accelerators and so probably represents "state of the art" for lithium ion engines. In future there's intent to use a pistonless system and electromagnetic pumping: Electromagnetic Pumps for Conductive-Propellant Feed Systems which should avoid some of the issues with trying to wrangle liquid lithium which has a depressing tendency to corrode mechanical systems. Note the thermocouples on all the valves, which aren't just there to melt frozen lithium in a cold system, but also to deliberately freeze liquid lithium in closed valves to help stop the stuff oozing out and ruining everything.
So, lithium is stored as a solid, melted and fed into a controlled supply cylinder, which then feeds it into a vapourisor, which then feeds into what looks to my thoroughly untrained eye like a more-or-less standard electrostatic gridded thruster, using a ring-cusp discharge chamber, though the diagram quality is rather low so I won't repeat it here.
This design (apparently dreamed up in the 80s, so it is surprisingly old!) was used in the NSTAR engine (which propelled Deep Space 1 and Dawn) and NEXT design. Those used xenon propellant, but the design apparently works well enough for anything that you can pump in as a gas, as it has been used with argon and lithium-fuelled test models, too. I found a 2005 dissertation Dicharge Plasma Processes of Ring-Cusp Ion Thrusters which has this handy diagram:
Gaseous propellant is blown into the discharge chamber through a hollow cathode (and a couple of other places). Electrons travelling between the cathode and the anode formed by the chamber wall ionise the propellant. Apparently a potential difference of a mere 25 volts is enough for this. A larger voltage drop (1000V would be required for 50000 seconds performance) between the chamber and the grid accelerates the positive ions out of the business end of the thruster.
As this seems to be a fairly standard ion thruster design, there's lots of other papers out there about it, such as this one: Ion Propulsion: An Enabling Technology For The Dawn Mission with similar diagrams and explanatory text, if more information was needed.