In short – no.
The itinerary receipt sent to your email address after payment serves as proof of purchase.
You don’t have to show it to anyone at the airport, as all your details are stored in the airline’s database, so your travel document is all you’ll need in order to check in.
To board the plane, you need a boarding pass, which you can get at the check‑in desk at the airport, at a self‑check‑in terminal, or by checking in online. In the last case, you’ll receive your boarding pass by email for you to save to your phone or print out.
This is roughly what an itinerary receipt (ticket) looks like.
If your ticket hasn’t arrived in your email inbox, check the Spam folder. If it’s nowhere to be found, contact the seller you booked through to get your ticket.