Is it possible to use Ultrasonic sensor HC-SR04 in continuous power-on mode
Yes, a 7805 would be acceptable for powering the HC-SR04. It only requires 15mA which is not much. The trick with voltage regulators is to not run too much power through them.

Find the max power dissipation of the 7805, if you can't find it then 250mW or 500mW might be a good place to start. 

Lets say you have 12V going into the regulator, that would be a 7V drop. At 15mA, this would be 7V*15mA = 100mW which would be ok (you can't go over the number or the 7805 will get too hot)

24V would be a 19V drop, and 19V*15mA would be 285mW which might be pushing it for some regulators.

Make sure you use input and output filter caps.