While a common enough form in general use, the use of the plural third-person pronoun with indefinite antecedent – for example, “Could the owner of the red Mazda please move their car” rather than “his or her car” – is by no means clearly correct. Authors as respected as Jane Austen have used it, but the confusion of number it represents can rankle.
The most obvious reason for the usage is the brevity afforded, especially when the pronoun is to be used a number of times. The OED has an entry for ‘they’ that reads: “Often used in reference to a singular noun made universal by every, any, no, etc., or applicable to one of either sex (= ‘he or she’).” It should also be borne in mind, however, that dictionaries don’t necessarily report on the ought, but rather the is of the use of words. For example, another entry the OED lists for the same word is as demonstrative determiner, as in, “They cows over in that there field”.
In support of the form, the quite common use in European languages of the second-person plural pronoun (French) and third-person plural (German) as an honorific at least presents something of a precedent. Also, the point seems finally to relate more to spoken than to written language. A very good discussion of the topic can be found at russinoff.com. The author raises some very interesting points, and brings a rigorous, and refreshing, formal logical analysis to bear on the subject. Well worth a look.