I have searched the tafsir for this verse and came up with this, please have a look.

"(And for such of your women as despair of menstruation) because of old age, (if ye doubt) about their waiting period, (their period (of waiting) shall be three months) upon which another man asked: “O Messenger of Allah! What about the waiting period of those who do not have menstruation because they are too young?” (along with those who have it not) because of young age, their waiting period is three months. Another man asked: “what is the waiting period for those women who are pregnant?” (And for those with child) i.e. those who are pregnant, (their period) their waiting period (shall be till they bring forth their burden) their child. (And whosoever keepeth his duty to Allah) and whoever fears Allah regarding what he commands him, (He maketh his course easy for him) He makes his matter easy; and it is also said this means: He will help him to worship Him well." Link

As you can see that Tafsir of Ibn Abbas even narrated an incident where some one asked Prophet Muhammad directly and He replied that these young girls if got divorced, there period of Iddah would be same. So from the time of Prophet Muhammad's time, its seems clear that girls were given in marriages, even prior to attaining the age of puberty.

Now one question may arise that, though its possible that young girls may get married prior to the age of puberty, how ever its not necessary for them to have sexual relationship with their Husband, well, I think even this case would not make sense if we read the other Quranic verses in this regard, which clearly commands to Muslims that Iddah is only applicable, meaning that a girl or woman is to observe Iddah ONLY if she got married and have sexual relationship with her husband, other wise there is no Iddah. Please have a look this verse;

"O you who believe! When you marry believing women, and then divorce them before you have sexual intercourse with them, no 'Iddah [divorce prescribed period, see (V.65:4)] have you to count in respect of them. So give them a present, and set them free i.e. divorce, in a handsome manner."

now as we can see that logical out come of these verses. In 65:4 you have mentioning of Iddah period of women and girls, one must assume that Iddah is only required after a woman get married and then divorced and Iddah is only to be observed when this women or girl got married and had sexual intercourse with Husband !

So my question once again, should be assume that Islam allow's marriages of young girls before the age of puberty ?

Peace and all the best