Do I need to have the My bpost app to be able to use a Parcel Locker?

It depends on the type of Parcel Locker. Parcel Lockers with a screen (most of them) can be used with or without the My bpost app.

Screenless Parcel Lockers can only be operated with the My bpost app. Your smartphone will communicate with the Parcel Locker via Bluetooth to open it. If you receive a parcel in a Parcel Locker, the e-mail announcing your parcel will tell you which type of locker your parcel will be delivered to. It also explains how the Parcel Locker works.