I try search "どゃでぇ" in Google just get almost nothing.
The どゃ portion
Notice that the や is basically the same font size as the ど. This is not どゃ, where the small ゃ is meant to indicate a palatalized glide attaching to the consonant of the previous mora, but rather どや.
I'm vaguely familiar with the adverb どやどや used to describe lots of people milling or rushing about; I wonder if it's used here to indicate the focus of the "camera" rushing towards the character.
Update: As explained by Chocolate in the comments, this どや is not from どやどや, but rather from the Kansai-ish どや that is equivalent to Kanto どうだ, as used in the expression どや顔【がお】. Literally "how's-that face", it's a bit like the English expression "how ya like (it / that / me) now", also used to indicate smug self-satisfaction.
The でぇ portion
I'm not sure about the でぇ portion. I suspect this is used to highlight the self-satisfied smug expression on the character's face, but that is only my intuition.
The でぇ portion appears to be the extended version of sentence-final で, the Kansai-ben equivalent to Kantō-ben emphatic sentence-final particle よ.
- For どや, see the 京ことば section of the Weblio entry.
- For で, see the second entry in the 大阪 section of the Weblio entry.
Any comments to clarify or correct the above would be most welcome.