After discovering that LED drivers are not the right solution for driving motors, I noticed that stepper motor driver boards use a ULN2003 motor driver chip. These are available on eBay for thin money, and I had a few lying around.
The ULN2003 can drive 500mA across 7 outputs, which is about 70mA per line. That's in line with what the motors need, and with PWM and non-constant activation this should be a good solution.
We've noticed that mounting the motors to duct tape - an easy, rapid prototype solution - doesn't work well due to the inherent stiffness of the tape. Any motor will vibrate a fair length of the tape, making it difficult for the user to distinguish one motor from another.
It's become clear that this project won't have a simple implementation, and we need to do some actual development and experimentation.
...which actually makes the project more interesting.
Dev kit for experimentation
So I made up a dev kit of sorts, with 7 motors on 14" wire wrap leads, so that we can experiment with motor position, different mounting fabrics, and so on.
It's also battery operated, because of course you never want to connect something to your body that also plugs into the wall.
Start with a ULN2003 on a small blox breakout
Add some headers left and right for the motors (bottom header) and control (top header)
Headers take wire wrap quite well. The common power lines are also pin 9 on the chip, so that works out well.
Add a big blob of hot glue to act as a strain relief. This will prevent (or at least, greatly reduce the chances of) the wires from breaking at the header interface.
Add a 3 volt supply and activation lead.
To use, connect the activation lead (one end shown on the blox) from V+ to any of the 7 unconnected header pins on the left, and the corresponding motor will activate. Short out several pins to V+ to activate several motors at once.