Do I need to install the Webp Express plugin for ShortPixel to work?

Hi Guna!

No, you don’t have to install any additional plugin. Please have a look at this article to know how to properly integrate LiteSpeed Cache and ShortPixel (in case you want to use LiteSpeed to deliver WebP): https://help.shortpixel.com/article/264-how-to-deliver-the-webps-generated-with-shortpixel-with-the-litespeed-cache-plugin

Have a good one!