This is definitely caused by the Deezer app. Only happens when I open Deezer and it sits in the background after closing it.

Hi @Rooleyrools it might only happen with the Deezer App but you need to change it in your general phone settings. Please search for the notification settings for the Deezer App - you can definitely turn it off 

Change deezer playback notification to silent

Thanks for sharing @Famous where did you come from? 

So I've got to choose between having the convenient playback in the drop down menu or having this app vibrate every single time I open, close, switch songs, etc? For how much I'm paying for it, there's got to be a better option. 

No. I have playback showing in the dropdown menu and have never had the app vibrate when starting, stopping or changing tracks, and I've been using Deezer on Android since 2016. One of your phone settings is causing this.