Is working in a children's clothing store considered unlawful (haram) if a committed young man is compelled to look at, smile at, and speak with mothers and girls (often immodestly dressed) for the purpose of making a sale?

Work is permissible, provided one lowers his gaze from women and does not engage in excessive conversation with them, except for what is necessary for buying and selling. If you adhere to these regulations, there is no blame upon you. However, it is better for an unmarried man to avoid environments that involve dealing with women.