Zener voltage is not stable
This is a very dangerous power supply, as it will be "hot" relative to earth/ground.  For safety, you really should use a power transformer to isolate the DC output from the AC.

Check the voltage on C2 when the relays are on. It probably drops below 7 volts, and your load doesn't leave enough current to keep the Zener conducting. 

The current drawn will cause varying voltage drops across R2, and also across C1 and R4.  You may have to increase the value of C1 to get enough voltage at C2 to keep the Zener diode conducting.