Subject pronoun or object pronoun in a particular case?
In general, subject pronouns are only used when they are subjects of finite verbs.
In this case, the daughter is not—she's the subject of a non-finite verb, more precisely a gerund. When the subject of a gerund is a pronoun, it is either an object pronoun or a possessive pronoun:
I don't like him doing this.
I don't like his doing this.
Possessive pronouns are more formal and the only ones considered ‘correct’ in traditional (= Latinate) grammar, but object pronouns are just as common, if not commoner.
So you can choose quite freely between object and possessive pronouns in your example here, but subject pronouns are not an option.
Even more commonly, though, condone takes a simple noun phrase, rather than a gerund phrase, as its object. So “Her mother condoned her illegal transactions” would be more common than “Her mother condoned her doing illegal transactions”.
Note: “doing illegal transaction” is not English. I'm not even entirely sure what you're trying to say—are you implying that the daughter was transferring money to/from someone she wasn't legally allowed to? Transactions should probably be plural, or if not then at least it must have a determiner, “an/the illegal transaction”. Do is also not commonly associated with transaction. Carry out is more commonly used.