That's because "chieste" agrees with "le", and it can only agree with the direct object, while "her" would be an indirect object. So in this case "le" is the direct object. You can use the clitic pronoun "le" as an indirect object but in this case it would be "chiesto", not "chieste". In that sentence there is no indirect object specified - you have no way of knowing who is being asked.
The translation doesn't seem right to me - I would translate "Certainly he will have asked for them". Here the verb "chiedere" means "to ask for".
he will have asked them = either lui gli avrà chiesto or lui avrà chiesto loro
he will have asked her = lui le avrà chiesto (note that here "chiesto" agrees in number and gender with the subject, while in "lui le avrà chieste", "chieste" agrees with the object "le")
"le" can be both an indirect (feminine singular) and direct (feminine plural) object pronoun. In this case it is a direct object pronoun that refers to something feminine and plural (e.g. "le sedie", "le magliette"), and you can see that because the "chieste" part of the verb changes accordingly, while it does not change when "le" is used as an indirect object pronoun: compare "lui le avrà chiesto" ("he will have asked her") and "lui gli avrà chiesto" ("he will have asked him"). It can be useful to remind that while in English you ask "for something", in Italian you ask "something" (direct object).
This is a very difficult part of Italian grammar! Clitic pronouns such as "le" and "gli" behave differently than "normal" pronouns such as "lui" and "lei". At which point of the tree is this topic presented?
As explained by several people above-
"le" only means her if it is an INDIRECT object. If it had been an indirect object here then chiestO would have been the participle--the participle only changes to agree with DIRECT OBJECTS--
but it was chiestE- which means "le" has to be a DIRECT object agreeing with the end of the participle. And in Italian, the DIRECT object of chiedere is the thing you ask for. The person you ask is an INDIRECT object.
He will have asked her- Le avra' chiesto
He will have asked for her (or a feminine it) - La avra' chiesta
He will have asked them - Avra' chiesto loro
He will have asked for them -Le avra' chieste or Li avra' chiesti