Are there any systems in the human body that work (sorta) like a pulley?
Sorta, yes. At least some extensor muscles (triceps, for example) for the major limbs have some sorta lika pulley action. Their tendons go over the elbow point and pull across it in a pulley-like action. It’s not a proper pulley, of course, but it does translate direction of motion by pulling a flexible “string” across a roundie bit.
The tendons of the hand also, but again, this is only like a simple pulley system, not a compound one.
What holds back a pulley system is that no animal has anything like a wheel inside them. (Well, maybe a dog does after eatting Tonka truck, but not usually.)
Two of the six muscles that move the eye (the inferior and superior obliques) run through “pulleys”. See here, for instance: