Older is definitely better, but there are other things you need to know about in the first few months after the disaster.
Firstly, lead-acid batteries self-discharge over a few months and once discharged, they sulphate and become scrap. So as soon as possible, you need to scavenge lots and lots of batteries and put them on float charging. How? Well, a solar panel generates ~250W at ~30V on a sunny day and ~25W on a cloudy one. If one knows a little electronics, a low-current precisely regulated voltage source is not hard to lash up. Or you could recharge them every month at higher current, using a scavenged DMM (digital multimeter) to work out when they are fully recharged. Or hack the mains-driven float charger in a battery distributor to work off solar panel DC instead.
Next, Tyres and other rubber components will rot in air. Much faster, if in the sun, and in the case of tyres if they go flat with the weight of a car on them. So you'd want to stockpile appropriate hoses and tyres, and store them under cover. A cave or a deep basement would be ideal (stable cool temperatures year-round).
Finally, fuel. It will probably last a year (for use in a vehicle without modern fuel injection) but it won't last a lifetime. Modern fuel-injected vehicles, most especially diesels, are highly sensitive to the quality of your fuel. Degraded fuel will destroy the injection system.
So to the choice of vehicle. I'd suggest two options. Firstly, light military vehicles (jeeps, land-rovers, humvees). Preferably, older. The military want their vehicles to run on whatever is available: dodgy diesel, kerosene, petrol, cooking oil, heating oil, and to heck with the clouds of smoke. If anything will run on decades-old fuel, an elderly mil-spec vehicle will.
Secondly, if you can find one: a car or bus that has been converted to run on LPG (Liquid Petroleum Gas). I expect that a tank of LPG (mix of propane and butane) will stay much the same until the tank fails and it vaporises. Centuries, probably. Such cars are rarities, but they do exist. The other snag is that LPG cannot be used for compression ignition. Such a car will depend on spark ignition, and therefore on having a battery.
Fork-lift trucks commonly run on LPG. Might it be feasible to hybridize a forklift truck and a petrol car using just hand tools and welding gear? Those prongs out the front might actually be useful for shifting debris off the roads.
If all the lead-acid batteries are scrap by the time you get on top of things, there may be alternatives. An AA NiMH rechargeable battery is not destroyed by going flat and is typically 2Ah. Eleven in series is ~13.2V. About sixty in parallel should be able to power a starter motor (briefly needs several hundred amps under effective short-circuit conditions). So you need to connect up an array of ~660 AA cells to replace a lead-acid battery. Maybe. I've never tried! The voltage is not exactly the same as a lead-acid battery, so the vehicle couldn't be trusted to charge them correctly without some electronic tweaking. But it would start and run minus the alternator, with a solar panel lash-up for charging the battery pack.
Or you might be able to find a NiFe battery. These are the ultimate rechargeable batteries for post-apocalyptic survivors. Overcharge them, undercharge them, short-circuit them, leave them to go flat and then years longer: they will survive it all. Because they weigh twice as much as Lead-Acid and because they go flat all by themselves in a mere few weeks, they are now rarities. The London Underground (railway) still uses them to power its overnight maintenance train locomotives, and if there are any pre-war lifts (elevators) still in use in their original form, they were used for regenerative start/stop operation.
Summary. Batteries, petrol/diesel fuel, rubber components: these are the things that you can't rely on scavenging, because they decompose quite rapidly whether in use or not. For the rest, if you can build a fleet of identical vehicles, then every part that breaks can be replaced from one of the others, for many decades. (The ultimate longevity of electronics is probably not yet known, which is another reason why old vehicles pre-dating mission-critical automotive electronics are best).