Do you have to be proficient in armour to be able to actually wear it?
No, you don't. However:
If you wear armor that you lack proficiency with, you have disadvantage on any ability check, saving throw, or attack roll that involves Strength or Dexterity, and you can’t cast spells.
(Player's Handbook, page 144)
The rules are designed so that it's not a good idea to regularly wear armour you're not proficient with, but they give you the option for when you really need the AC more than you need your abilities, attacks, or spells.