I'm making a campaign setting in the post-apocalypse of the real world. I have tribes of primitive humans...

Spiders, Centipedes, Leeches, Scorpions, Cockroaches, Snakes, Crocodilians, Sharks, Poisonous frogs, And that's all I can think of right now.
Also everywhere.

A burgeoning civilization of intelligent cars.

Predators aren't really a problem for humans, even in extremely low-tech situations. Bears, wolves, tigers etc. don't seek out groups of humans as prey, and are easily scared off if they do come into contact with a group of people. Consider that the stone age peoples hunted mammoths, 8 ton monsters with natural weapons and inch thick skin, armed with nothing but wooden spears - humans are just that much better off in regards to teamwork, intelligence and tool use than animals are.

Your best bet is some kind of supernatural being that for contrived reasons are dangerous to and specifically target humans. Zombies are a classic.

With giant cats as their predators.

Like Jurassic park, but with vehicles. The players will be running for their lives from packs of intelligent mopeds and one large bus.

if you don't want magic monsters then you're kind of stuck with really bold or dangerous versions of modern animals (bears/wolves/tigers etc) or other humans, maybe roving bands of crazed cannibals or whatever.

Or maybe some big dangerous monsters were genetically engineered before the apocalypse and have escaped from their confinement and started breeding.
Basically I'm saying they should be dinosaurs.

This sounds fun too

Yes.

They dwell in the ruined cities of the world, supping on the last remnants of gasoline and electricity, and boiling victims into diesel.

en.wikipedia.org/wiki/Coywolf

A super rabies that makes even deer aggressively dangerous monsters.

Mutie scum

I always thought it would be fun to consider the long term effects of GMOs in a fantastical way.

A new radical method of gene modification sweeps through the scientific community, revolutionizing the biological sciences and making gene modding a common, almost trendy practice.

Basically, the gene mods that were being thrown around willy nilly had unforeseen consequences for later generations. Over successive generations, the genes that were spliced in Start to assert themselves. And unfortunately, this gene modding trend came about just as the apocalypse hit. Now, without the technology to correct these changes, plants, animals and people start to mutate, giving rise to any number of horrifying chimera beasties,(owlbear anyone?) carnivorous and or mobile plants, And savage mutated beast people.

You already have the greatest monster. Man.

Pale milky-eyed humans who are adapted to subterranean life.
They're not necessarily hostile, but do dislike and frequently steal from the surface-dwellers. They also worship strange gods and sometimes partake in human sacrifice.

Have you thought about what caused the apocalypse? Because you know that's the first thing your players are going to try to find out

>Survival rate: 0%

Pleiastocene fauna, after the mammophant program worked better than anyone thought it would things took off at a break neck pace and soon everywhere had its own pleistocene park filled to the brim with all sorts of ancient critters, that had a bad habit of getting loose and running wild.

>What are good options?
>I don't want magic stuff
Unfortunately, it's either that or a "science" explanation that might as well be magic if you want big threatening "monsters". Realistically we're quickly wiping out large animals and if there were to be an "apocalypse" of some sort then if it didn't finish them off humans would in desperate need without access to traditional modern food production or governing bodies able to track and restrict poaching of vulnerable populations.

If you're willing to compromise some and put in the effort to portray things as threatening besides just making them big and spiky you could have interesting threats from smaller creatures and swarms. Plague-bearing and radioactive pests don't necessarily make for the best traditional "fights" though if you're looking for a combat focus.

Alien tourists. You're beneath notice, they came to see the triffids bloom, but they're disruptive and understanding where you can safely stand when they transmaterialize around is beyond you.

shannara.wikia.com/wiki/Creeper

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