Is it possible to make an LED turn on with an HC-SR04 without using an arduino? Only when an object is near it?
It's not impossible. See http://www.buildcircuit.com/how-to-test-dyp-me007-ultrasonic-range-finder-using-ne555-and-multimeter/



But it won't work like you want without a needlessly complex circuit. The HC-SR04 or similar ultrasonic range finders have a fairly small output timing, that tells you how far something is. If nothing is near, the output is high for 32ms, indicating a "not in range" state. Hooking up an led to this won't only turn it on when something is near. And it depends on how often you pulse the trigger. As you see in the video on the link, the output is fast enough that the multimeter averages it out. An LED would do the same, essentially PWM.

Your two real options is use a micro controller which is dead simple (and there are plenty of projects where all the code is already done for you), or use a different sensor, like a basic PIR sensor module, which work basically how you want it to, light up when something is near. Well, moves near in front of it.