However, as the official statement of Nabisco, the manufacturer of Oreo, suggests, "Many Oreo products are suitable for vegans but may include cross-contaminants of milk."
While Nabisco doesn't specify the reason, the statement likely implies that Oreos are produced on equipment that also handles milk, exposing the cookies to the dairy product.
To cover itself legally, Nabisco doesn’t promote Oreos as perfectly dairy-free, although its ingredients don’t include dairy, and any dairy that does get in is almost negligible.
Vegans or people allergic to even a trace of dairy should avoid Oreos. Lactose-intolerant people, however, can handle the small amounts of dairy that may be there in the cookies.