The sentence with correct pronoun and antecedent agreement is:
While the neighbors were walking to the store, they dropped their money and the wind blew it away.
In this sentence, "they" correctly refers to "the neighbors," and "their" also corresponds appropriately to "the neighbors."
The other sentences contain issues with pronoun-antecedent agreement:
- "Carol wanted to go outside, but it was raining and she could not find our rain jacket." - Correct, but repeated.
- "When the teachers arrived at school, she saw the students writing with chalk." - Incorrect, as "teachers" is plural and should have a plural pronoun rather than "she."
- "Because the brothers were tired, he forgot to shut the gate and the dog ran out." - Incorrect, as "brothers" is plural and should have "they" instead of "he."