Agree with Dave it might just need a tune up since it fires right up. I'm assuming it's an old school coil and capacitor ignition system, right? We have an old school propane Allis Chalmers 4k that we bought for scrap value many, many years ago because the surplus equipment dealer couldn't get it started. My dad inspected the machine and then had me try to start it while he played with plug wires and distributor cap. After a couple mins of cranking without even a single cylinder firing, my dad handed the guy a check for the asking price. The guy kinda laughed and asked if we needed it to be forked onto our trailer. At that point my dad pulled a new 98 cent condenser cap out of his pocket and hooked it up in under a min. Forklift fired right up and we drove it onto the trailer. I'll never forget the look on the dealer's face...
If it's the coil/condenser system, I recommend replacing plugs, gently filing/sanding the contact or point, make sure you have a solid power feed lines through coil and strong ground connection with capacitor. Sometimes the little wires that power and connect the coil and condenser get old and brittle from vibration and the wires are broken internally near the connections. You can measure a good voltage or ground when the machine is off but then when you start it, it starts missing because the wires are shaking and causing hit/miss on supplying power or ground. And make sure the condenser is fully grounded through it's little mounting screw. Plus a good cleaning and drying out the distributor cap to ensure there is no carbon, dust, or moisture buildup is a very easy way to see if that helps as a first wave of tune up..
If not, then recommend replacing condenser and coil and making sure you have a strong power and ground connection to the distributor to see if that helps. If not, then replace the spark plug and condenser wires as they may be broken internally also.
Also, I'm purposely not discussing the propane fuel system as you said it fires right up but it could be an issue too. Does your lift idle without missing? Or does it miss at idle and high rev (no load)? Or miss only under load? Understanding a bit more when it's missing might provide clue if it's starving for propane. There are random inline filters that might be clogged but that's a low probability. I personally wouldn't touch the fuel system until I know the ignition system is working perfectly as it does start (gets fuel) but runs rough (ignition or starving for fuel). We never touched the propane supply system on our fork so I'm kinda biased it's the ignition giving you issues.
BTW- Our little fork is still the main workhorse for moving stuff in/out of the shop. Almost anytime it doesn't start or run without coughing is because of the coil or condenser has a bad power or ground connection.
