So the Warlock Pact of the Chain says
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. (PHB 107)
And the Find Familiar spell states
Finally, when you cast a spell with a range of touch, your familiar can deliver the spell as if it had cast the spell. Your familiar must be within 100 feet of you, and it must use its reaction to deliver the spell when you cast it. If the spell requires an attack roll, you use your attack modifier for the roll. (PHB 240)
Now, the sprite entry says
Invisibility, The sprite turns invisible until it attacks or casts a spell, or until its concentration ends. Anything the invisible sprite is carrying or wearing is invisible as long as it remains in contact with the sprite. (PHB 310 & MM 283)
I added the Monster Manual and Player's Handbook both as references because the stat entries are identical, and neither of them ever state that a Sprite has spells it can cast.
Combining the "familiar can deliver the spell as if it had cast the spell" with the Sprite's invisibility saying "invisible until it… casts a spell", I'm sure a DM could easily rule that it then turns visible.
However, the Imp and Quasit also turn invisible but their entries are
Invisibility. the imp turns invisible until it attacks or until its concentration ends. Anything the invisible imp is carrying or wearing is invisible as long as it remains in contact with the imp. (PHB 306)
Invisibility, the quasit turns invisible until it attacks or uses Scare, or until its concentration ends. Anything the invisible quasit is carrying or wearing is invisible as long as it remains in contact with the quasit. (PHB 309)
(Their Monster Manual entries only differ by saying "concentration ends (as if concentrating on a spell)."
Going off of these, it would appear that being used as a touch spell conduit "as if it had cast the spell" doesn't make the Imp or Quasit lose their invisibility. Is this the case?
Also, if a specific spell requires an attack roll, does using them for it count as them making an attack for the purpose of losing the invisibility?
Best Answer
Channeling a spell through a familiar ends their invisibility
As you've already determined, the line about "...delivering the spell as if it had cast it" is enough to trigger invisibility ending on a spell cast.
Any of them would lose their invisibility after delivering a spell
Quasits and Imps can't normally cast spells so their invisibility blocks don't list "or casts a spell" as an end condition. Since all the invisibility stat blocks do list all of the offensive actions the monsters are normally capable of as ending invisibility it's pretty easy to figure out what should happen here. So all of these monsters will end their invisibility if you channel a spell through them.
A strict RAW ruling would disagree, but 5e just isn't written tightly enough to support that kind of ruling.