Relay not turning on when supply comes from LDO Regulator
Recommended operating limits 7805 datasheet:




You  can see that V_min is 7V, while your Battery voltage is 6V!


Why do you need an LDO ?


In your circuit configuration it makes absolutely no sense! Further, if you want to switch the relais, use a Pmos or Nmos transistor instead of a BJT, depending if you prefer a low-side or high-side switch.
Since a relay consists of a coil, don't forget to include a suppression diode in parallel to your relais. 
You should also read the datasheet of your relais about max/min ratings. 







simulate this circuit &ndash; Schematic created using CircuitLab


Be aware of stablity issues!


An inductor makes a regulator more unstable!