Can I power my shift register with a 12V external power supply and connect only the data, strobe and clear pins to the Arduino digital pins? I'm using a CD4094 shift register to power my LEDs along with a ULN2003A darlington transistor array. I want to hook up the LEDs to an external power supply of 12V (common anode). 

The cathodes of little groups of LEDs will be connected to the ULN2003A which is connected to the shift register which will get its data from the arduino (the digital pins which i assume work at 5V).

My question is, if I connect the Vdd pin of the shift register to the 12V external power supply, will it still function as intended through the Arduino?
No. CD4k devices reference their high and low voltages to a (mostly) fixed ratio of Vdd, so there will be no way for the Arduino to operate the device.

Consider using a TPIC6C595 instead, which combines the shift register and the high-voltage driver into a single device.