I'm not sure on the servos. They might have to have some juice in order to crank a gumball machine. Any recommendations?
My favorite servos are from the Robotis Dynamixel line. They are a bit fragile if you run them at high voltages (16V) but at 12V they run fine. They start at $40 or so for the AX-12A, and go up from there pretty steeply. I think the AX-12A could do it.
The AX-12A can actually do full-turns, as opposed to the 180 degrees of a hobby RC servo, which is nice.
You control these servos using a serial port, rather than a PWM pin, so interfacing with an Arduino is slightly harder, though.
Another option is to just use a geared brushed DC motor with some kind of sensor to know when it should stop running. That's what you can drive with the motor shield you linked to.
It looks as if most of those pieces are compatible with each other. I don't know if there will be some interference between the actual pins needed by the motor shield, the keypad, and the I2C for the clock chip -- there are only so many pins available on the Arduino, and some pins are "special."
Finally, yes, Arduino is very much "a thing" these days :-)