My garage lift only works if the light switch is on!
It sounds like they ran power to the motor and light via the light switch instead of directly to the motor which would have a much smaller, separate set of low voltage wires running to the garage door button.  If you flip the light switch the motor and light now have power which then makes the garage door opening button able to do its job.

The only way you can confirm this is by pulling out the light switch off the wall and looking to see how many sets of wires are in the box. I really can't imagine they've somehow split the power to the light and motor for the garage door at the destination though.  If so, then they aren't utilizing that functionality by having one light switch control it all.  This sounds like they wanted to do something cool, didn't think it through very well, did the work and then realized it wasn't going to do what they intended and left it.  Either that or a bad wiring job.

If you're sure this is how it's controlled and there are no other factors involved, and there is one set of wires running to the single gang light switch, then no, you would not be able to split the power to the light and motor without isolating the power to the light alone and then running power to the motor on a new line that would bypass the light switch.  You'd could check the motors docs though, there may be a way to split power at the motor already which would at least make the job a bit easier.  Bathroom fans with built in lights are doing this to make splitting the fan and the light to separate switches much easier now a days.  You just have to run separate *edit (control power wires) to the unit.

In my opinion, I'd remove the switch from the equation completely, and just make that a junction box with a cover and power the door the way it was intended.