A vampire that can't return to its resting place becomes a vampiric mist.

As detailed in Mordenkainen's Tome of Foes (which came out just a month after I posted the question) on page 246 (italic emphasis mine):

In billowing clouds of fog lurk vampiric mists, the wretched remnants of vampires that were prevented from finding rest. Indistinguishable from the mists they lurk within, they strike unseen and undetected to bleed their victims dry.

Former Vampires. Vampiric mists, sometimes called crimson mists, are all that remain of vampires who couldn't return to their burial places after being defeated or suffering some mishap. Denied the restorative power of these places, the vampires' bodies dissolve into mist. The transformation strips the intelligence and personality from them until only an unholy, insatiable thirst for blood remains.

I think the focus here is on the second part - "after being defeated or suffering some mishap". If the vampire decides to chill with his vampire buddy next town and therefore skips a day's sleep, I don't think he'll instantly die, just like a human won't die from skipping a night's sleep. However, if adventurers reduce him to 0 HP, triggering the vampire's Misty Escape, and he can't go to his coffin - then he does indeed die (as specified in the Misty Escape feature, except it doesn't mention Vampiric Mists).

Vampires suffer the same consequences from skipping sleep as regular humanoids.

Contrary to the average "Undead Nature", vampires don't ignore their needs for sleep, drink, and food. They only ignore their need for air, as specified in the MM on page 295:

Undead Nature. Neither a vampire nor a vampire spawn requires air.

This means that, since nothing states otherwise, they require sleep as much as your average humanoid (elves aside).

Indeed, the lore text states that they "must rest by day". However, I believe this only means they have to rest in the first place, and they can do so only during the day, not at night (though I would probably ignore the latter part in games I DMed). If vampires couldn't possibly be awake during the day, that would make it pretty goddamn easy for adventurers to kill them.
Vampires are CR13, i.e. a "medium challenge for players of level 13". At this level, the adventurers have access to spells like Teleport or Plane Shift, which means they can port out of the vampire's resting place if they're in a tight spot, and they can scry and port back once they've recovered. If the vampire were necessarily asleep from 8 am to 8 pm (roughly), then that would be way too easy.

"Warforged, roll out!"

The Integrated Tool feature says (emphasis mine):

Choose a tool you're proficient with. It becomes integrated into your body, and you double your proficiency bonus for any ability checks you make with it. You must have your hands free to use this tool.

So RAW this means any tool you are proficient with is eligible. So the question becomes whether something is a tool.

For the RAW, I would assume anything listed in the Tools table in the PHB (p. 154) counts, as well as anything listed in similar list in later books.

The last line of the Tools table lists "Vehicles (land or water)", so by RAW they should qualify. This is further empowered by the fact that the Sailor (PHB p. 139) and Soldier (PHB p. 140) list "vehicles (water)" and "vehicles (land)" as Tool Proficiencies respectively.

So yes, RAW vehicles could be integrated into a Warforged Envoy.

Beyond RAW

As a DM, I would rein this in a bit for instance by only allowing person-sized vehicles. So you can integrate a small rowboat or a cart or sled (PHB p. 157) - something you'd maybe able to carry 2 other people in, but not really self-powered.

Having said that, I could very easily see other DMs saying no to the idea. So if you want to actually use this feature this way, discuss it with your DM first.

So no, you could not be an integrated Air Galleon (though this might be an idea for a Knight Rider one-shot...)