「Verb in Volitional Form + って + なる」
has been an extremely common colloquial phrase for, I would say, the last decade or two. It might get completely outdated in the near future, or it might last long enough to make its way into Genki. Common words and phrases come and go rather quickly in Japanese.
If I understand it correctly, this sentence structure means "何々をしたくなる".
Exactly. It means 「Verb したいっていう(気持{きも}ち/話{はなし})になる」.
Is the saying "意向形ってなる" fairly informal?
Yes, it is very informal. We will call you out if you use it in any formal situation. It sounds light and casual and it is mostly used by the younger generations.
And, can I replace "って" with other particle, such as "と" or "に"?
Good question. Using "と" would be "possible" but since it is an informal phrase, 「と」 would not sound very natural.
To use 「と」 and still sound natural, you might need to use the "traditional" phrase 「~~たいという気持ちになる」 instead.
Using 「に」 would be incorrect; It just will not be grammatical.