The walls were designed to be hard to climb
As mentioned in the comments on the question, the scenes from the films of orcs climbing everywhere in Moria were from Peter Jackson's imagination. There are several breeds of orcs; those from the Mines of Moria are smaller and might be skilled at climbing cave walls (as Jackson imagined), while the Uruk-hai (a large part of the force attacking Helm's Deep) are larger and perhaps no so good at climbing.
It is reasonable to assume that defensive walls are designed to be as smooth as possible and difficult to climb. We know that the Númenóreans who built Orthanc were very skilled at this. The Ents, who had no trouble destroying much of the Ring of Isengard, are defeated by the stones of Orthanc.
Many of the Ents were hurling themselves against the Orthanc-rock; but that defeated them. It is very smooth and hard. Some
wizardry is in it, perhaps, older and stronger than Saruman’s. Anyway they could not get a grip on it, or make a crack in it; and they were bruising and wounding themselves against it.
The Lord of the Rings Book Three, Chapter 9: Flotsam and Jetsam
The walls of Minas Tirith were built by the same people and were considered to be unusually strong and smooth.
At first men laughed and did not greatly fear such devices. For the main wall of the City was of great height and marvellous thickness, built ere the power and craft of Númenor waned in exile; and its outward face was like to the Tower of Orthanc, hard and dark and smooth, unconquerable by steel or fire, unbreakable except by some convulsion that would rend the very earth on which it stood.
The Lord of the Rings Book Five, Chapter 4: The Siege of Gondor
Helm's Deep was also built in earlier days by the Númenóreans or their descendants.
At Helm’s Gate, before the mouth of the Deep, there was a heel of rock thrust outward by the northern cliff. There upon its spur stood high walls of ancient stone, and within them was a lofty tower. Men said that in the far-off days of the glory of Gondor the sea-kings had built here this fastness with the hands of giants.
The Lord of the Rings Book Three, Chapter 7: Helm's Deep
If I were sending a horde of orcs to attack a fortress designed by these people, I would send some ladders.