How can I turn LPG from liquid to gas?
You don't really need to do anything to get LPG vapour. It is a gas a room temperature and pressure so as long as you just want the clean gas not mixed with air (which would be dangerous as the mixture will be flammable and potentially explosive), all you need to do is allow the output of the canister to vent into a balloon.

LPG is mostly propane and butane which liquefy under mild pressure. Butane will liquefy at about 220 kPa at normal room temperature (this is a little over twice atmospheric pressure). This is what happens in a typical cigarette lighter like the one illustrated. There is no "wick" involved just a valve that allows the gas to vent to the atmosphere and a flint to create a spark and ignite it (which you can avoid doing by careful handling of the lighter activating just the valve and not the flint). Large cylinders are similar and it might be possible to vent directly into a balloon just by carefully controlling the exit valve on the cylinder.

It is possible that standard supplies of LPG for household use also contain stinky additives to make leaks easy to detect, but otherwise the gas should be similar to that found in a cigarette lighter.