The question was asked on the SSSF forum several years ago. Since then I’ve kept a lookout for wildlife films containing penguins. With the “Blue Planet” and “Spy in the huddle” series now on, I can at last report tentative first observations, not yet enough to distinguish different answers for different species.
Consider first how a human lying on belly stands up. I stand up by either pushing with both arms to get into a kneeling position, or pushing with one hand to get sideways bringing knees up and then either getting sideways into a cross-legged sitting position or twist knees to get into kneeling position. Then up one leg at a time from kneeling position with added push from arms.
Penguins don’t stand up from belly down this way. But they have a large variety of ways that they do stand up from belly down.
1) Rocket upwards out of the sea directly onto their feet.
2) With feet forwards stand up directly from belly down position on level ground. Emperors do this.
3) as 2 but using one or both flippers to get added lift.
4) as 2 but facing into an upwards slope for added lift.
5) as 2 but using the outrush of a wave for added lift.
6) accelerate forwards up slope for added lift.
7) on level ground jump forward into a run with head down.
8) when on uneven ground push sideways with legs like a vice to lever up.
Most of these I’ve only seen one penguin do – wildlife documentaries seldom film penguins standing up from a belly down position.