They don't have a diaphragm.
My tig has a similar pump, it was easy to see how it[why it didn't work in my case] worked it was just valves and
the soldenoid did the pumping from what I remember.
I bought the replacement from a german company. But there were loads of them for sale in the UK[described as for coffee machines], finding them
was just a matter of getting the naming right, a "solenoid pump thingie" wasn't correct.
I wouldn't have thought anything wants to be greasy, I would try putting it back together with it all clean and dry.
they have them in the karcher carpet shampoo machine's . solenoid pump. the piston - valve slides backwards and forwards and it has a spring and a one way valve inside . no lubrication needed if its not pressurising the barrel and piston wears normaly then u/s
There is similar used as fuel pumps for mini diggers and other small engine driven plant . Most are plastic pistons over a steel core in a plastic body so you can pump oils fuels of water /cutting fluid .
Inside there is a small circuit board that turns DC into pulsed DC to operate the elecectro- magnetic coil that makes it pump . If left turned on & it runs dry the coil often burns out .
Find them on eBay as " 12 volt electric fuel pump piston type " or similar eg eBay # 233433943090 approx. £9.00