Can I use 3v CR2032 to power multiple vibration motors?
Obviously, with a discharge current of 0.2 mA, there's no way you can power even one motor that draws up to 70 mA, let alone 6.

Look into small Lithium Ion batteries, something like this can produce a sustained 80 mA, which will power one of your vibration motors. It could maybe run all 6 briefly, but I would go bigger just to be safe.

For what it's worth, I would try to not run all 6 at the same time if you can get away with just doing one at a time. You also could PWM the motors so they don't draw their full complement of current if you need to limit that. 

Also, if you're going for maximum power or efficiency, connecting these motors in series probably won't give you the results you want. Go parallel or separately powered. 

Use a transistor of some sort to switch the power—the output pins of an Arduino won't be enough to drive them.