In a comedy, this idea might fly (pun). But there are some reasons it won't work seriously.
1 ) Precise amounts of energy are needed to lift precise amounts of payload. This is solvable through math and trial and error, but then you would need to carefully weigh each occupant before liftoff.
2 ) Humans are fragile. Even if the elevator was very sturdy, didn't vibrate, was unimpeded on its ascent, and grabbed on tight at the top of the journey, there would always be a hard jolt at the beginning. John Stapp's tests showed that 42 Gs can be tolerated for very short times - I will leave the calculation of the maximum height achievable with a human body for posterity.
3 ) Fuel quality is low, if you assume biologically made gas in a leather bag. Let's assume that this society has figured out the ideal mix of methane with atmosphere in a sealed bag to explode. It still can't be stored at high pressure in a leather bag.
4 ) Lastly and most importantly, an unconfined explosion is mostly wasted. Guns work because the explosion of the gunpowder is channeled down a tube and not allowed to escape. In order for your idea to work, the shaft would need to be very well sealed, and not let gas out the sides or up past the car.
If the shaft is not durable metal, it will quickly be damaged by the explosions and let gas escape.
I suggest hydraulics instead.