Regulating VAC transformer voltage
Well I have a AC go DC circuit. The transformer is rated input is 220VAC and then outputs about 15VAC which is then rectified then smooth capacitor then voltage regulator of 12V. 


Since you have a 220Vac-to-15Vac transformer (so n = 14.67), the secondary voltage will be about 13.5Vac if the input voltage drops to 200Vac.


  I am asking because I have a 12V Voltage Regulator and when the VAC voltage is below 220V the voltage regulator does not output 12V.


If you rectify the secondary and use a sufficiently large smoothing capacitor then you'll have a DC bus voltage of at least VDC = 13.5 x 1.4 = 18.9VDC which is still enough to regulate to 12VDC.

You didn't provide any info about the load but I think the load draws high current. So the smoothing capacitor cannot filter out the excessive ripple thus the DC bus voltage drops a lot. If you are using a linear regulator (e.g. LM7812) then the input voltage of the regulator should be at least 2V higher than the desired output. So, if the DC bus voltage (i.e. the voltage across the smoothing capacitor) drops to 13VDC or lower (due to the unsufficient smoothing capacitor) then you cannot get 12VDC output.

I'd suggest you to use either paralleled smoothing capacitors or a DC-DC buck converter (better choice) instead of regulating the line voltage.