Animal is not a creature type
Going pure RAW, the centaur is a fey creature and not a beast. Now, 'animal' isn't a game term and speaking from a biological point of view, we're all animals (assuming mythical fey count as part of the animal kingdom) - but if we look at most draft animals with their 'beast' type in 5e, then we can separate out the fey from the beast when talking about animals.
With this in mind, it doesn't seem like it would act as a mount in this way with regard to pulling a vehicle.
Instances of Animal seem to be talking about Beasts
The term animal is used a bit sparingly and never in context with a specific creature type, but it does seem to be used when talking about beasts and not other creature types.
Appendix D: Creature Statistics states:
SPELLS AND CLASS FEATURES ALLOW CHARACTERS to transform into animals, summon creatures to serve as familiars, and create undead. Statistics for such creatures are grouped in this appendix for your convenience.
Every creature that isn't summonable as a familiar or undead listed in appendix D is of the beast type.
The Mule stat block also gives the following Beast of Burden trait:
The mule is considered to be a Large animal for the purpose of determining its carrying capacity.
Again, this suggests that beast and animal may be of the same.
We can also look at the skill Animal Handling which is separated from dealing with other creature types. While a DM could allow a Animal Handling check against any other creature, the intent seems to be that it is for (PHB, 178):
When there is any question
whether you can calm down a domesticated animal, keep a mount from getting spooked, or intuit an animal’s intentions, the DM might call for a Wisdom (Animal
Handling) check. You also make a Wisdom (Animal Handling) check to control your mount when you attempt a risky maneuver.
Having said that, I don't think it'd be gamebreaking to allow a centaur to pull a vehicle and act as a mount in this way. The primary strength of the centaur is in it's lower body and it does have an Equine Build.
The problem is more of an issue when it comes time for combat. I can't find any rules around the time it takes to hook a draft animal up to a vehicle, but it's not negligible. A DM will need to come up with a houserule for this timing, but it's most likely that the time required is more than a minute - which would be problematic if combat begins while they're acting as a draft animal.
Additionally, you would need a customized harness for the centaur.
Finally, it's unclear if an intelligent centaur would be okay with being used as a draft animal. That will also come into (role)play.