TL;DR: We just don't know.
The "car" suddenly sputters to a halt. He quickly blames the issue on
Steelerfan the battery:
Rick: Oh, great. Oh, boy.
Morty: W-what's wrong, Rick? Is it the quantum carburetor or something?
Rick: "Quantum carburetor"? Jesus, Morty. You can't just add a [burps]-- Sci-Fi word to a car word and hope it means something... Huh, looks like something's wrong with the microverse battery.
A bit later, inside the battery microverse, Rick discovers the cause:
Rick: Huh, this isn't right. This pipe's supposed to be sending 20 terawatts of juice up to the engine, Morty. [Computer beeps] Instead we've got... zero?
What this means:
This wouldn't make sense with a normal car, because while the engine of a car is running, it is recharging the battery. As the engine runs, the battery is allowing energy spikes and shorts to travel harmlessly to the "ground", but it isn't actually making the car run. The battery starts the engine, but doesn't directly keep the engine running. The alternator usually has enough power to keep everything working without the battery. Removing the battery from a running car can do terrible things to the vehicle, but it usually won't directly make the engine stop running (although it may do so indirectly, due to the aforementioned terrible things).
So the fact that Rick's "car" immediately stalls when the battery dies means that it is clearly not a normal car. This was already pretty obvious, since it flies and travels through space and cuts people to pieces with lasers. Does it have a secondary power source? We don't know. Could the alternator keep the defense mechanisms working? We don't know, but that would be handy - Rick is an interdimensional criminal being hunted by law enforcement and shady alien mercenaries alike, and he's smart enough to think about what might happen if his pursuers came upon him while he was fixing the
Quantum Carburetor... something or other.
What we can say:
The attention to detail and continuity that Harmon and Roiland have maintained on the show so far suggests that this isn't a goof, so we can probably assume that the "car" has some sort of backup power that is sufficient to keep the defense systems working, but can't/doesn't keep the "car" running.
In short, the engine wasn't running, but at least some parts of the vehicle still had enough power to maintain functionality for at least as long as Rick and Morty's trip into the microverse battery took.