this autois refers to the things to be used and goes with chresetai. chresetai autois - he will use them, ὁ ἀεὶ μάλιστα δεόμενος. the one who is at the time in need, υμων of you: the one among you who is at the time in great need, will use them.
Yes Chris, Const.Philo puts it together for you. Literally it goes “the (one) of you being most in need (ὑμῶν ὁ μάλιστα δεόμενος) will use them (χρήσεται αὐτοῖς, i.e. will have the use of them—this verb takes the dative).”
αει with the attributive participle (ὁ ἀεὶ μάλιστα δεόμενος) has its regular idiomatic meaning in such phrases, not “always” but more like “on any given occasion.” (The stock example is ὁ ἀεὶ βασιλεύων.)
αυτοῖς refers to the gifts that Cyrus has been given.
χρησεται is third person singular, not plural. In LSJ, the entry for this verb in the sense of “use,” is χράομαι, not χράω. (This is confusing, misleading and frustrating, to be sure.) Under χράομαι II, LSJ indicates that in the sense of of “use,” this verb takes a dative object.