As with Bethpage Black, carts, aka buggies when we move overseas, are not allowed at any time. As noted by the R&A in squashing John Daly’s request for a cart.
From Alistair Tait’s Golfweek report, that also suggests the R&A had other reasons besides steeper-than-normal dunes:
“We believe that walking the course is an integral part of the Championship and is central to the tradition of links golf which is synonymous with The Open. We must also ensure that, as far as possible, the challenge is the same for all players in the field.
“The terrain at Royal Portrush is not suited to buggies and indeed the club itself does not permit their use. We have a serious concern that some parts of the course, where there are severe slopes and swales, would be inaccessible.”