What should my divorced sister do, who is unemployed and has a daughter whose expenses are high, and her mother is the one who spends on the household, while her ex-husband provides monthly support for the daughter, and she doubts the source of these funds?

Knowledge of the source of funds is not a prerequisite for their lawful use, nor is it obligatory to inquire about their source. Indeed, it is impermissible to do so if based merely on suspicion, as this entails شرعية [religious] prohibitions, including: excessive meticulousness (al-tanattu'), and causing distress to the person being questioned. Allah Almighty has commanded us not to ask about things which, if made manifest to you, would trouble you. Therefore, you must accept the money and refrain from inquiring about its source.