The technology you will use for the construction of your cloud chamber depend on what you want to do with it : research, challenge of construction, educational purpose, permanent exhibit… There is 4 technologies of cloud chamber where the characteristics can be summarized below from my experience :
If you don’t have access to dry ice, I recommend you to build a thermoelectric cloud chamber. This will cost a few hundred $. There is plenty of tutorial on youtube to build one. My first build was a thermoelectric aircooled cloud chamber with a sensitive surface of 12 x 4 cm, pictured below. On my alternate youtube chain, you will find the videos related to this little machine. You can download an ancient pdf guide to build this machine, but you have to considerably improve the design, I wrote this 9 years ago.
With an aircooled machine you can’t put too much peltier because of the amount of heat generated. If you want a bigger cooled surface, you need more peltier and this will get you to a watercooled, thermoelectric cloud chamber.