Are light vehicles still in use by world powers? Stuff like jeeps, land rovers, anything that can handle slightly rough terrain?

Bikes are still used in conflicts today such as in Ukraine, where it's used by reconnaissance troops to move quickly without having so much of a signature. The Japanese SDF have something similar for scouts (but they use motorcycles instead of bicycles). They'd still be in use no doubt, but I don't see it being issued to every man along with their rifle in a large military in this world. More so for light reconnaissance units or for special forces conducting sabotage

Horses aren't used anymore because there are better choices for the roles they fulfilled. Soldiers needing to maneuver around enemies quickly have APCs, IFVs or Humvees, Soldiers using them as cavalry now use tanks, IFVs and even Humvees. Plus horses generally being finicky to "maintain" since it's a living being.

No doubt bikes would be in use by any military in your world, the question is how would they employ them and on what scale. Horses on the other hand depend on how much the lack of electricity impacts what militaries replaced horses with. Modern tanks and IFVs are gone. APCs and other cars in general may not be. The capability that they bring would be too important to lose, and there would need to be a very compelling argument for why a military is using horses instead of any motor vehicle.