Ideas for precisely moving a small object in 2 dimensions?
For a really small X-Y table, a pair of the tiny stepper motors used for camera lens focusing would serve well: 

These steppers have a 6 mm slide span, and allow better than 0.01mm precision:



These slightly larger steppers have an 18 mm slide span, and precision of around 0.025 mm.



In each case, microstepping allows the precision to be improved by a factor of 8 to 32, depending on the specific driver and drive mode chosen. 

If the 22 mm overall length of the larger stepper is a concern, with a little research, a 9 or 10 mm slide span stepper, in the same 5 to 7 mm motor size, can surely be found.

Using a StepStick microstepping driver, I have been able to get practically vibration-free operation from the smaller stepper mentioned above. Results should be similar for other such lens focusing steppers.

This will permit an X-Y table to be fabricated with a footprint comparable to the movement dimensions mentioned in the question, i.e. around 15 mm x 22 mm including motors, with the motors placed under the table.

As a matter of fact, an actual X-Y table would not be required, since the object to be moved is small: A single-axis sliding base served by the first stepper would work, with the device being moved along the perpendicular axis directly by the slider tab of the second stepper. 



Bonus: If microstepping is not required, these steppers work even when driven directly off an atTINY45 microcontroller's GPIO pins, since their required operating current is in the 3 to 8 mA range. One just needs to add flyback diodes to protect the pins.