I think you would need a thousand pounds of fuel for the flight.
the massive Saturn V rockets are designed to lift the fuel off and get the think up.
the efficiency is so bad , like 1%, that for every once of payload, you need 1,000 pounds of fuel
helium balloon is he cheap way to get really high.
the Paris gun... or some such, could send a projectile into space.
this is something you will need a lot of backing with, and permits as it will cross every conceivable aviation territory.
your vertical shot means it could wind up in other countries. most governments want to know well in advance about such things.