No Purana has a date, because no Purana is one text. Each is a compilation that was added to, rearranged and locally recensed over centuries, and manuscripts of the same Purana from different regions can differ so far that they share little beyond a title. The Skanda Purana is the extreme case: a Nepalese palm-leaf manuscript of the ninth century preserves a version substantially unlike the printed vulgate.
The datable spread runs roughly from the third or fourth century CE, for the oldest strata of the Vayu, Brahmanda, Markandeya and Matsya, through to material still being added in the sixteenth. The Bhavishya Purana, whose name means "future", contains passages describing Mughal and British rule, and in some printed editions Queen Victoria, which tells you precisely how open the genre stayed. Dating individual passages is possible; dating a Purana as a whole usually is not.
All of them claim greater antiquity. The tradition holds that Vyasa compiled the Puranas as he compiled the Vedas and the Mahabharata, and each text is framed as ancient speech relayed through a chain of narrators, commonly the bard Lomaharshana or his son Ugrashravas reciting to sages assembled in the Naimisha forest. That framing is doing real work: it borrows the epic's authority and asserts continuity with the Veda for material that is often quite new.
The relation to the Veda is the point of the genre. A Purana is smriti, human in composition, and beneath shruti in formal rank, but it may be recited to and by anyone, including women and those excluded from Vedic study, and it is often described within the tradition as the fifth Veda. Formal subordination, practical primacy.