Learn Croatian: Nemojte jahati vrućom pustinjom! - Do not ride through the hot desert!

This is the bare Instrumental of path (the prosecutive). When you travel along, over, or through a continuous space, Croatian often uses the Instrumental case entirely on its own to show the route of motion.

Could I use the preposition kroz (through) here instead?

Yes! You can absolutely say kroz vruću pustinju using the Accusative case. It means the exact same thing. The bare Instrumental (vrućom pustinjom) is just a slightly more stylistic or traditional way to express the path you are taking.

Does the verb jahati work for riding vehicles as well as animals?

No, jahati is strictly used for riding animals, like horses or camels. If you were warning someone not to ride a bicycle or drive a car through the desert, you would need to use a verb like voziti se.

Why does the sentence use Nemojte jahati rather than Ne jašite for 'Do not ride'?

Both are grammatically correct negative commands for the plural or formal 'you'. However, using the auxiliary nemojte plus the infinitive (jahati) is the standard, slightly softer way to give a negative command. Saying Ne jašite! sounds much more abrupt and forceful.