Learn Hardcore Japanese: denwabangou wo matigaezuni okutta node, henzi ga sugu kita. - Because I sent the phone number without mistakes, the reply came quickly.

電話番号を marks the object “phone number.”
間違えずに is the adverbial form of the negative verb 間違える (“to make a mistake”), built by taking the 未然形 stem 間違え and adding ずに, meaning “without making a mistake.”
送った is the past tense of 送る (“to send”).
Altogether it means “(I) sent the phone number without making any mistakes.”

Why is ずに used instead of ないで in 間違えずに?

Both 〜ずに and 〜ないで can mean “without doing ~.”
– 〜ずに is somewhat more formal or written.
– 〜ないで is more colloquial.
In this sentence, 間違えずに sounds concise and slightly more formal, fitting a written or polite context.

What role does ので play in 間違えずに送ったので、返事がすぐ来た?

ので is a conjunction meaning “because” or “since,” used to link a cause or reason to a result.
Here, 間違えずに送ったので means “because (I) sent it without mistakes,” and it introduces the outcome 返事がすぐ来た (“the reply came quickly”).

How is 返事が来た different from 返事をくれた?