"has" or "have"? As in "X requires that Y have ..." versus "X requires that Y has ..."

Your first example is in the subjunctive mood; the second is in the indicative. The subjunctive refers to "a mood of verbs expressing what is imagined or wished or possible." That's taken directly from NOAD, and you may be interested in their usage note:

USAGE These sentences all contain a verb in the subjunctive mood;: … if I were you;: the report recommends that he face the tribunal;: it is important that they be aware of the provisions of the act. The subjunctive is used to express situations that are hypothetical or not yet realized and is typically used for what is imagined, hoped for, demanded, or expected. In English, the subjunctive mood is fairly uncommon (esp. in comparison with other languages such as French and Spanish), mainly because most of the functions of the subjunctive are covered by modal verbs such as might, could, and should. In fact, in English the subjunctive is often indistinguishable from the ordinary indicative mood since its form in most contexts is identical. It is distinctive only in the third person singular, where the normal indicative -s ending is absent ( : he face rather than : he faces in the example above), and in the verb ‘to be’ ( : I were rather than : I was , and : they be rather than : they are in the examples above). In modern English, the subjunctive mood still exists but is regarded in many contexts as optional. Use of the subjunctive tends to convey a more formal tone, but there are few people who would regard its absence as actually wrong. Today, it survives mostly in fixed expressions, as in : be that as it may;: far be it from me;: as it were ;: lest we forget ;: God help you;: perish the thought; and : come what may.

The takeaway from this is that you can use either; if you wish to use a formal register, the subjunctive may suit you fine. If not, there is nothing preventing you from using the indicative.

Use have. It's subjunctive mood, which in English is conveyed with the infinitive (uninflected) form of the verb. As Robusto's answer mentions, it's often considered optional, so you can make the stylistic choice of has over have, but I'd say have is preferable in formal writing.