An error crept in, 名前無し there should have really been 名無し.
で indicates a way something is done. Thus コート無しで means literally "in a coatless fashion". But you don't have to always use で, for example, 名前無し means nameless or anonymous. Also, it's an noun-like suffix, and you use it with の in adjectival context、 like in 名無しの人。 For fun factor, it seems that -無し was the negative suffix in classical Japanese, equivalent of modern-day -ない. So although it's an equivalent of a suffix inflected like i-adjectives (形容詞), it behaves like a noun today.