It’s been bugging me for a while, but recently I have a character who would actually take them.
There are no vehicle spells in 5e D&D. No “make the thing go faster”, no “teleport this vehicle to a different place”, no “make this vehicle move silently and leave no trace”, no “create a vehicle with these limitations”, no “this vehicle can fly for X time, nada.
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I'd look toward summoning spells.
To benefit from
a faster cart, supposedly horse drawn, replace the draft horses. Phanton Steed have the same stat as a riding horse (so they can't carry as much as draft horse, but they can still carry a cart and 480 lb of equipment) but with a speed instead of 100 feet, twice the speed of a real horse. Nothing prevents you from putting a harness on the steed, but nothing in the spell description says you can, so, depending on GM's ruling, you could have a spell effectively doubling your cart speed for an hour for a 3rd level spell.
Teleport this vehicle is done through Teleportation Circle (5th level slot). It creates a 10-feet wide circle, and your average
Prairie Shooner was 4 feet wide according to the Britannica. It should fit easily inside the circle.
To
make your cart leave no trace, the easiest thing is to make it hover above the ground. Tenser's floating disk can carry a loaded cart on top of it and make it hover 1 meter above the ground. Silent, no trace. According to the spell description "The disk is immobile while you are within 20 feet of it. If you move more than 20 feet away from it, the disk follows you so that it remains within 20 feet of you.". If nothing can push the disk, then you can't just hop on the disk and have it be moved. So you'll have to conjure a phantom steed to carry you (with a flying speed of 100 feet) and by RAW the disk will follow you as long as you don't cross impassable terrain because it remains withing 20 feet of you in any case. Tenser's spell last for one hour, no concentration, phantom steed is the same : they are made to work together.
To have a quicker ship... It might require some DM's adjudication. The listed speed of 2 knots for sailing ships is on the very low end of medieval transportation, an average speed much lower than the top speed to account for the fact that most of the time, you couldn't get the wind right. Using Control Winds (5th level slot), you guarentee a strong wind (20 mph) in just the right direction regardless of the local weather condition. It's enough to maintain top speed for the duration of the spell (15ish knots for a longbot, 10ish for a medieval merchant ship). That or you could bind a water elemental (using conjure elemental and planar binding) to push the ship, but the net effect would depends on GM's ruling. By
To create a vehicle, you'd need an
ad hoc spell. Fabricate would require the wizard to be proficient since it's a specialized craftsmanship and a Large item would be a very small ship indeed. I'd simply adjudicate that
Mordainkainen's Phantom Palanquin would be an up-level variant of the Phantom Steed. starting at level 5 (group effect instead of single-target spell sounds good for a 2 level difference). If you just wanted a cart without it being self-propelling, I'd follow dave2008's advice and use the UA spell "Find Vehicle" (not technomagic) to create a fey, fiendish or celestial vehicle lasting 8 hour, but otherwise totally normal, with an upcast possibility to carry more people.
To make a vehicle fly... no Floating Disk gimmick. Note that boots of flight (4h hour of flight daily) are a mere uncommon magic item. Horseshoes of flight and wheels of flight (to make the cart stay horizontal) shoudn't be more expensive. If you're in a dire situation, Animate Object (5th level) can turn your Huge cart into a Huge construct with a fly speed of 30 ft but only for one minute. Then again, your best bet would be to summon an Air Elemental or two (depending on the required carrying capacity) and Planar Bind them to grapple your cart and carry it with a speed of 45 ft for 24h.
Since the DMG is lacking in balancing advice regarding utility spell versus combat-oriented spell, I guess that homebrew spells doing the same thing that can be achieved by manipulating combat-oriented spells should have a lower level, or a longer duration. For travel-oriented spells, duration is the main problem. If you don't want to cart to be able to attack by itself, extending its duration through animate object to 8 hours shouldn't be a problem considering it's a 5th level spell...