「さっそく頼みがあるんだけど」
I'd like you to do me a favour right now
「そうじゃそうじゃ。なんでも言っとくれぃ」
My guess was that it's a contraction of 言ってくれ, the imperative of 言ってくる.
The person asking the favour is sat next to the other person so くる as 'come' makes no sense. That leaves me with 'start to say anything', which also makes no sense.
I also thought っとく could be a contraction of っておく (I've seen that before) but that didn't help at all either.
It's 言う + て + おくれ + い. The ておくれ part is contracted to とくれ.
There are two points:
おくれ
This is a mostly outdated word that functionally equals くれ. Etymologically it's honorific prefix お- + くれ. くれ is the imperative of くれる, not くる as you suggested (Both くれる and くる are irregular; くれる > くれ and くる > こい). We already have a good answer for its usage so please see: Is おくれ interchangeable with くれ?
V-て + くれる form roughly tells that the action is favorable to the speaker. Again see: Use of くれる with 信じる.
い
Although written in small letter as if some kind of suffix in this passage, it's a dictionary approved particle that means similar to よ. This word also has mostly died out, so you don't have many chances to see it outside stereotypical parlance in fiction. It reminds people of brisk speech of traditional town-dwellers, namely 江戸っ子.
To conclude:
なんでも言っとくれぃ
Ask/tell me anything! (Don't hold back, as I'm happy to help)
