The "dictionary" form of:
「そんなワケねーだろ、このめでてぇ日{ひ}に。」
would be:
「そんなわけないだろう、このめでたい日に。」
The original phrase is in the Kanto tough guy speech. Occuring twice in that short phrase is the "famous" あい-to-えぇ/えー switch. 「ない」 became 「ねー」 and 「めでたい」 became 「めでてぇ」. This phenomenon has been discussed many times here on SE, so I will just leave it at that.
「ワケ/わけ/訳{わけ}」 here is somewhat difficult to translate all by itself, but if I had to, I would go with "reason" or "ground".
I would, instead, suggest that you remember 「そんなわけ(は/が)ない」 as a set phrase meaning "said no one ever" if I could go somewhat slangish. To use a more traditional TL, it would be "there is no way (that is true/possible)"
This phrase is used very heavily among native speakers. You will often hear the more slangy form 「なわけねーだろ」 as 「そんな」 is often contracted to just 「な」 in very informal speech.
On a side note I assume めでてぇ is a casual form of omedetou?
No, it is not. As I stated above, it is the same as 「めでたい」.
So, the phrase in question means:
"There's no way that's possible on a happy day like this!"
Provide more context if this does not sound right.