Just one of those situations where French doesn’t always need a preposition “manger/mâcher la bouche ouverte/(fermé)”. My understanding is it would not be “wrong” to use ‘avec’ but it is not the way it is commonly said in France.

The same is heard in English - “he eats/ chews mouth open” - although less frequently.

In English you can do something similar, although you'd use a comma:

He chewed, mouth open, before he took a sip of water.