For local webp you need to install webp in the os if you did not already do so.
eg
apt install webp libwebp-dev
or compile from source, I have always had no issues with webp on Hestia
You can check if you have webp enabled in php with
php --ri gd
output of command
GD Support => enabled
GD headers Version => 2.3.0
GD library Version => 2.3.0
FreeType Support => enabled
FreeType Linkage => with freetype
GIF Read Support => enabled
GIF Create Support => enabled
JPEG Support => enabled
PNG Support => enabled
WBMP Support => enabled
XPM Support => enabled
XBM Support => enabled
WebP Support => enabled
BMP Support => enabled
TGA Read Support => enabled