I am attempting to repair a lemax Halloween decoration for my wife. The power adapter says 9 v output but there are 2 motors inside the decoration that shows 12 v. Does this mean the adapter is rated too small to operate the motors?
A motor in itself can run at any voltage up to its rated voltage (and beyond provided a heat sink is added). The performance of the motor in terms of speed VS torque, and therefore the [speed;torque] operating point, will be decreased accordingly though. If the stall torque (maximum torque at 0 rpm) is smaller than or equal to the load torque (due to static friction first), the motor will not start at all and there will be no operating point.
Imagine your motor has a torque constant of 1 N.m/A, and a winding resistance of 10Ohms. Let the static friction of the load prevent any movement up to 0.5N.m. The motor will barely start turning when the motor torque goes past 0.5N.m, which requires a current of 0.5A. 0.5A flows through 10Ohms when 5V are applied.
Applying more than 5V will make the motor speed up. However, as it speeds up (unless it operates in vacuum) viscous friction will increase, requiring more and more torque. In addition, as it speeds up the effective voltage applied, and therefore the motor torque, will decrease due to a counteracting "back electromotive force" (which is a voltage, not a force) induced. The rotor rpm will stabilise when motor torque balances load torque. That's the operating point.
In a nutshell, there is nothing wrong with running a 12V motor on anything below 12V, but it will be weaker, and therefore slower, or even may fail to start depending on the load.
Your 12V motors are overkill, but 12V motors are very common, so they must have been included in the design for their cost effectiveness and used under conditions that are less demanding than what they've been built for.
Driving motors is not a very precise art (unless you want it to be!). They can begin turning (With no load) way lower voltage than their "rated" or "designed" drive voltage.
A 6V motor may begin turning at 2V for example.
The issue is, as Andy Aka is hinting to in his comment, voltage relates to how much torque the motor can apply, which also ends up relating to velocity. The motors sound like they arent doing much, so 9V is obviously good enough to operate the motors at the required speed and torque.