I see a way to make it possible with a bit of adapting, but my answer is still theoretical:
Some basic digital alarm clocks use a "divide by 60" counter to convert the 60Hz wall signal (in Canada/US) to a 1Hz signal to count seconds. It's a crude design since the frequency may be 57Hz or 63Hz, but oddly it averages to 60Hz pretty well. So if you open up a digital clock, you may find these counter ICs. If the clock is primitive and uses Dual Inline Package ICs (through-hole chips with about 7 pins on top and 7 on bottom), you can look up the IC number and find if they are counter ICs. Now if you are so lucky to find this to be true, then you need to know the condition on all the bits that correspond to the desired alarm time. Feed all those pins into an AND gate, which drives the digital signal of a motor driver (digital signal cannot drive the motor directly).
After writing this and rereading your reply, try taking a look on ebay or AliExpress. Some people create custom circuits for $5-$10 that are timers and have the display integrated into it. It is probably easier to use timers than clocks, but good luck at any rate.