Tapi is shortened from "Tapioca Tea" (i.e. Boba Tea or Bubble Tea).
This was chosen as the Buzzword for 2018 for JC (Josei Chuugakusei - Middle School Girls) and JK (Josei Kousei - High School Girls) by AMF (Some sort of Fashion Magazine, IDK.)
What was most likely used in the manga was the verb-ified form, タピる (Tapiru).
You wanna be a cool school girl like Nagatoro and friends?
[ol]
Just take a common place noun like "タピオカティー" (Tapioca Tea)
Damn that's too long, let's shorten that bitch to two syllables: "タピ" (Tapi). Much better!
Shit it's a noun! We'd have to use other parts of grammar to describe/modify it and that makes talking about it hard to do because you gotta add things like verbs and adjectives to it and that takes up precious oxygen in each breath I take and I'm not breathing anymore than I HAVE to.
I know! Let's make it a verb. Slap a る on there and we can conjugate this thing like fuck! タピる (To "Tapi")!
Thank god! Now I don't have to say タピオカティーを飲みたい。 (I want to drink Tapioca Tea.) just タピりたい。 (I want to "Tapi".).[/ol]