Anyone can use a mount or vehicle, there are no restrictions based on proficiencies.
Ability (skill) checks only come into play when the DM decides it. Generally it's reasonable to assume that most characters in a medieval setting have some knowledge of mounts and vehicles.
(DMG p. 237) Is a task so easy and free of conflict and stress that there should be no chance of failure?
And later on:
(DMG p.237) With enough attempts and enough time, a character should eventually succeed at the task. ... However, no amount of repeating the check allows a character to turn an impossible task into a successful one.
However, when trying to do something fancy, or to prevent a mount from spooking, or say your character has never ridden a horse before, then an ability check to Animal Handling or Vehicles (Land) may be called for at the DM's discretion.
Note that these can be called with different attributes; you may need Dexterity (Animal Handling) to stay on a bucking mount, or Wisdom (Animal Handling) to calm a frightened horse. (See PHB p. 175 Variant: Skills With Different Abilities)