In Yoga and Pilates, there is a pushup posture. I've heard it called many things: high-to-low pushup, chataranga, Pilates pushup, and four-limbed staff pose.
The point to keep in mind is how it's performed, regardless of what you want to call it.
The mistake I often see is when students allow their elbow to bow out away from the body, more like a military pushup. In both Yoga and Pilates, the elbows are encouraged to hug the ribcage while the fingers point forward with the crown of the head.
Making this adjustment may mean that you don't lower down as far as you do in a military pushup. There's no shame in that! I've been practicing Yoga for over 5 years and there are days when I only lower an inch or two in my pushup. Every practice will be different and that's the joy of it.
For more detailed instructions, and to see more photos, visit this demonstration.