Pickling without Sterilization - Is It Safe?
The recipe appears to be close to a standard Summer Pickle recipe.
It's about a 9% brine with 3.75% acetic acid, which should effectively discourage human pathogens.
It is not a fermented product.
They're meant to be refrigerated for at most a few months, not stored on some dusty basement shelf.
Refrigerated properly, I'd eat them without concern.