Not naturally, but it might be possible if you try hard enough.
Let's look at how a thunderstorm actually forms. Like any cloud, it forms when air cools below the dew point -- which typically happens when warm, moist air rises and cools (recall that the temperature drops as altitude increases). The condensing water releases latent heat, letting it rise further into cooler regions of the atmosphere, and grows higher. For a thundercloud, the limit is the tropopause -- the point at which temperature no longer decreases with height. This is the 12km number @RancidCrab gives, but it's not actually a requirement that it grows this high; it's just how things work out on earth.
What distinguishes a thundercloud from just a big cloud is, well, the thunder. This comes from a charge differential between it and the ground. The exact nature of this is still poorly understood, but the consensus is that it has to do with the formation of graupel (chunks of ice). In a thundercloud, it's (a) cold enough that graupel can form and (b) there's enough turbulence that small bits of ice can be lifted up and fall back down again repeatedly without falling out of the cloud.
In an O'Neill Cylinder, your temperature gradients are wrong: temperature is probably relatively constant or maybe even increasing as you get closer to the artificial sun. Also, as others have mentioned, you want one parcel of air to be hotter than its neighbors so it can rise up and form a cloud.
If you want lightning anyway:
- Add some heater in the ground for uneven heating. Maybe add some extra humidity for good measure. This will get you cloud
- Design some mechanism that will give you very steep temperature gradients. Perhaps the artificial sun only provides light, and heat only comes from the ground, and you have a cooling mechanism inside the sun?
- Try to encourage turbulence within the cloud. The temperature gradient will have helped with this, but you might be able to help it along by increasing wind shear: have fans at different altitudes that blow at different speeds in different directions, or whatever other sci-fi method you want. If you're clever, you can control wind currents within the cylinder just with strategic uneven heating of the ground. You don't have much vertical space to play with though, so your vertically circulating currents need to be strong.
Edit I just realized that for clouds to form nicely you want your environmental lapse rate (how fast air cools with altitude) to be between the wet and dry adiabatic rates on average (~3.5C and 10C respectively). On earth, it's typically about 6.5C/km. If you have a lapse rate of 9.5C/km and a ground temperature of 0C, it should be be just about possible to make things work in a 4km radius cylinder, but you'd really benefit from making it larger.
- I've ignored the effect of pressure, since that wasn't specified in the question
- I'm not sure what the changing "gravity" (from the spinning) does to buoyancy due to temperature differences at different altitudes. I'm assuming it cancels itself out, but I could absolutely be wrong.