Like Lawrence and others mentioned in their comments, it's commonly just called climbing.
to go up or ascend, especially by using the hands and feet or feet only
However, in rock climbing, there are various techniques. The type of climb you describe is known as the mantle or mantel
Mantle (n): A transfer of motion from pulling to pushing while rock climbing.
A helpful way to visualize a mantle is to picture yourself exiting the side of a pool. Before you leap out of the water, you have your hands on the poolside edge in a downward-pulling position. Then you jump up, engaging your arms and planting a foot on the ground. You then use your shoulders to push yourself the rest of the way up to a standing position.
Another method for flatter (not rounded) ledges is to think about it like exiting the deep end of the pool: Use your feet to spring your hips up over the lip, leading with your chest to get your weight over your hands. Straighten the arms, gently bring a toe up to the lip, and stand up slowly.
Depending on the context of where you want to use the word, it is unlikely that the ordinary layman or man in the street will have heard about the rock climbing technique.
Perhaps you could use scale as an alternative?
to climb up something steep, such as a cliff or wall:
He scaled a steep cliff beside the river.
to climb to or over the top of a high steep object such as a mountain or a wall
Student protesters scaled an 8-foot fence to enter the Embassy grounds.
to climb something with difficulty, using your hands and feet
clamber up/over/into etc: I clambered up the ladder into the hay loft.
After I had posted my answer Kris pointed out in a comment that there is a Wiktionary definition for mantel as follows:
mantel (third-person singular simple present mantels, present participle mantelling or (US) manteling, simple past and past participle mantelled or (US) manteled)
(climbing) To surmount a ledge by pushing down on the ledge to bring up the body.