Oh, no my little pony. I have a @_@... with it. It´s enough that we have a shadowheart ingame.
Now seriously, you can pick an unicorn as your patron as a celestial warlock. I do not know if it would let you mount it, but there are plenty of puns about your "horny" patron that you can do.
Mechanically speaking, a unicorn is a CR5 creature with legendary actions, so it would be very powerful and may require to tone it down. https://roll20.net/compendium/dnd5e/Unicorn#content
Either way, RAW you can only get this as your familiar,
You gain the service of a familiar, a spirit that takes an animal form you choose: bat, cat, crab, frog (toad), hawk, lizard, octopus, owl, poisonous snake, fish (quipper), rat, raven, sea horse, spider, or weasel.
Special Pact of the Chain:
When you cast the spell, you can choose one of the normal forms for your familiar or one of the following special forms: imp, pseudodragon, quasit, or sprite.
Ed: I forgot, following Volo´s book (if your DM allows it, with I do for example) you can add a cranium rat, a crawling claw, a gazer,... I do not know if Larian will include something outside the PHB.
but you can get it from another source. i.e-> You do not need to summon one to get a mount. Usually you can try handle animal, buy it, convince one to join you, even using spells like "Awaken" if you´re a bard or druid, etc.