Never try to meet the enemy on terms where they have a natural advantage. Change the terms.
Chariots are never going to match cavalry, and there's a reason that as soon as horses were bred that were strong enough to carry humans and equipment, chariots lost their predominance in warfare. Riders are more maneuverable and have greater flexibility in where they can be deployed. While cavalry isn't great over broken ground, they're a lot better than chariots who require really flat open terrain in order to be deployed. To use a simplistic example, dig a ditch a meter wide and half a meter deep, with steep sides. A horse and rider, so long as they aren't weighed down in full armour plate, and the horse has been reasonably trained, can easily jump it. It would be an impenetrable barrier to a chariot with reasonably-sized wheels. Or if you don't want to dig a ditch, make a small stone wall. A horse and rider could jump it...they might even be able to step over it. Chariots would come to a complete halt (or be smashed into pieces hitting it, depending on how stupid the drivers are).
So why would your four-armed people even bother to try? It would be like, to use an extreme example, mer-people trying to fight on land against humans. Sure, you might think of ways of doing it, but why would they even want to? The humans are always going to have the advantage. Make them come to the water.
You've got four-armed people. Assuming the four arms are equal in capability, say like the Green Martians from the Barsoom series, then you've got a monstrous capability in heavy infantry. Four arms means you can carry twice as many pikes, or pikes plus shields, or pikes plus swords, or pikes plus shields plus swords, or pikes plus bows, or bows plus shields, or so on and so forth. Let the humans runs around with their little horses, and see how effective they are going up against an infantry square where the front rank effectively impervious, with twice the overlapping shield capability of a Roman legion or Greek phalanx, while also carrying as many pikes per unit as a human formation could (one shield per arm, two arms on pikes), the second rank is carrying all pikes, so doubling what a human formation could do, and with greater size presumably heavier/sturdier pikes, backed up by a third line also carrying all pikes. And then you back that up with missile troops. Can they fire two bows at once? If not, well, they they could carry a shield as well to give them protection as they fire (which has traditionally been a weakness for archers).
No cavalry is going to break that line.
The biggest weakness in linear formations in the pre-firearms era has been the enemy appearing on the flanks because the troops are unprotected, but think about it: with four arms this isn't the case. The troops on the flanks can carry an additional shield held on the outside, which gives them protection from that angle, while the next file in can carry a pike pointing forward and a pike held off to the side. If they are charged from the side, the line might be weaker than the front, but it's pretty much as strong as a human front line would be.
tl;dr version: if you have people with different physical capabilities than ordinary humans, they aren't going to fight like ordinary humans.