"climb onto" is correct and usable in written English.
You can use it when you want to express the action of getting onto something. For example, "He climbed onto the roof to take a look around."
I climb onto Hibbah's big raft, in which everyone paddles.
It was another hundred yards or so before there was anything safe to climb onto.
The insurgents explode homemade bombs, climb onto school buses and strafe children with gunfire.
Frail elderly individuals would find it difficult to climb onto a nonadjustable exam table.
They wait to climb onto the giant Olympic rings sculpture and pose for more pictures.
They still have to climb onto rooftops or scale a pole to check wires.
In another they climb onto scaffolding, employing the bars for acrobatic moves and added sound effects.
Dale wandered off and came back occasionally to climb onto his father's lap.
Captain Sullenberger beckoned him to climb onto a life raft at the main cabin door.
She laughs when she sees the beds, and the two of them climb onto one.
My son wasn't the only one eager to climb onto a Vespa.
