I've come across both できんでえ and ~てやんでえ today but I'm not sure what they mean or what their origin is. There's some information relating to ~てやんでえ online which seems to point to it being a question ("what did you say?") in casual speech:
「保護と言えば、いつかこんなこと言っていた者もいたよ。『俺ら、前のことが分っちゃったもんで会社に入っている保険屋から断わらちゃったい。 ほんだもんでガン保険に入ることができんでえ。 そんな馬鹿な話あるけえなあ。 ガンが一番恐いのは俺たちなのになあ』って。 全くだよ」
Is that right?
Edit: The conversation is taking place between two men from 焼津市 and it's a real conversation that took place about 30 years ago if that helps at all.