You will want to bring cling film for the wax pot and it will need to be one with a lid. Clean it the second you are finished and wrap it if it’s still runny or you have a particularly bumpy ride to go on (eg country lanes).
Otherwise you should be good with just a shopper bag to carry it in the car between other stuff holding it up so it doesn’t knock over and if it does the car is protected.
I found a laptop backpack was brilliant. It was my husbands. Tons of pockets and opens down the middle so you can put towels, Pedi bowl filled with tools wrapped in more towels for cleaning. One side for used after, wipe clean, has front pockets for easy access to cotton/sticks/strips etc.
Basically something that has compartments so you can find everything or get clear zip bags so you can find it all. Once the towels are used and you are trying to get out quickly it never ends up perfect and tends to get crammed in so something with a bit of extra space would be useful and a spare carrier bag for emergencies (and wet stuff!).