The mounted combat rules only apply when you are mounted
Here's the introductory text for the mounted combat rules:
Controlling a Mount. While you're mounted, you have two options. You can either control the mount or allow it to act independently.
The rules specify how you can become mounted:
Once during your move, you can mount a creature that is within 5 feet of you or dismount.
So these rules only apply when you are mounted on a creature, via the mechanism provided - i.e. you pull yourself onto it and sit on it. Not when you are standing on a chariot a creature is pulling.
This also means the Mounted Combatant feat would not apply:
Mounted Combatant. You are a dangerous foe to face while mounted. While you are mounted and aren’t incapacitated, you gain the following benefits...
You get no benefits from this feat while not mounted.
The vehicle rules are minimal
The vehicle rules only specify how much an animal can pull:
An animal pulling a carriage, cart, chariot, sled, or wagon can move weight up to five times its base carrying capacity, including the weight of the vehicle. If multiple animals pull the same vehicle, they can add their carrying capacity together.
And that they have special proficiencies:
Vehicle Proficiency. If you have proficiency with a certain kind of vehicle (land or water), you can add your proficiency bonus to any check you make to control that kind of vehicle in difficult circumstances.
These rules do not specify precisely how combat on a chariot would work - so you will have to talk to your DM or make a ruling if you are the DM.
How I would rule
Personally, I would rule this situation as follows:
The mount dragging the chariot acts independently (and thus its actions are not restricted). A normal mount may do things the Paladin does not want - but this is mostly mitigated in this case by the bond specified by the Find Steed spell.
A chariot will have reins - the Paladin can hold these in one or both hands to control the mount and the chariot (restricting its actions as if he were controlling it while mounted). He might want to use these if he had a non-Find Steed mount, or if he wanted to control the chariot through a difficult area. I would probably give disadvantage to any ability checks to control the mount or the chariot if he held the reins one handed.
Side note: do my rulings make chariot use strictly inferior to mounted combat?
There are several reasons why you might use a chariot rather than ride a horse.
- Some chariots might have bladed wheels, allowing the chariot itself to become a dangerous weapon.
- More people can ride in a chariot than a horse can carry (see "five times carrying capacity" above).
- The chariot could be used as cover.