I have a player who has a Calistrian bard, and is looking for a way to aquire a giant wasp companion or familiar, or some type of relationship. Does anyone know of a mechanical way to achieve the result, or a similar one?
The easiest way to do this within the rules is the Eldritch Heritage feat, selecting the Arcane Bloodline. As long as you're comfortable taking a skill focus in any knowledge, your bard should qualify.
You can get an animal companion the same way, although it's a bit more of a stretch. The Sylvan variant of the Fey bloodline grants an animal companion. You'd need the skill focus to be in knowledge (nature) specifically for this.
If you are happy with the character having such a companion, don't worry about finding a mechanical reason, just design an adventure where the character can gain such a companion as the reward.