I'm trying to find relics that influence or are influenced by Charisma, but I cannot seem to find any.
What are the magic items that use Charisma or boost Charisma?
The Tome of Leadership and Influence, when read, permanently boosts Charisma and maximum Charisma by 2.
The Ioun Stone (Leadership) boosts your Charisma by 2, to a maximum of 20, while it orbits your head.
The Star card (from the Deck of Many Things) can boost your Charisma by 2, to a maximum of 24.
As far as I can determine, there are no other items that affect your Charisma.
Items that are affected by your Charisma is more complicated. There are a few magical items whose use involves making a Charisma check or a Charisma saving throw, but this isn't really using your Charisma as such. However, if you are a Charisma-based spellcaster, every item that mentions "your spellcasting ability" is referring to your Charisma. There are far too many of these to list.
Probably the most common use for charisma in items is with sentient weapons. Charisma checks are used when the item doesn't agree with its usage by the wielder and can try to (essentially) charm the wielder if the wielder can't come to an agreement.