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" refers correctly to "the neighbors," which is plural, maintaining the correct agreement.
The other sentences have issues with pronoun and antecedent agreement:
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"When the teachers arrived at school, she saw the students writing with chalk." (Here, "she" incorrectly refers to "teachers," which is plural.)
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"Because the brothers were tired, he forgot to shut the gate and the dog ran out." (Here, "he" incorrectly refers to "brothers," which is plural.)
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"Carol wanted to go outside, but it was raining and she could not find our rain jacket." (The pronoun "our" does not match with the singular "she.")