[Chris is me]

You can use a little surgical tubing, right? That's fairly grippy. Prototype that if you want to try something other than zip ties.

You can get various different FTC legal zip ties of different stiffnesses, try stuff out. You want enough force that it will propel the balls, enough flexibility so that you can keep the helix running with balls at the top without forcing them through the shooter mechanism, and enough stiffness that balls will not roll out the bottom. If you've achieved this, maybe you don't need to look for more solutions.

This is assuming you're using a helix that goes "up". If you're using one that goes "down", test and see if you need anything other than gravity. You might not.

As for how to power it, whatever works best for your design. In FTC I preferred to use gears whenever possible (granted this was vex). They all get the job done.