I am designing a creature that is quadrupedal but can run after prey on its hind legs. What kind of hip bone would allow for the excess stress to be put on it during bipedal locomotion? The animal is roughly the size of a mountain lion and weighs on average 100 pounds. It is a predatory carnivore. The planet it inhabits in very similar to Earth. It has an identical atmosphere that is 23% oxygen.
For vertical facultative-bipeds (Adapted primarily for walking on four legs, but able to walk on two legs) the only real consideration they have is where the center of gravity.
Bipeds need the center of gravity directly over the feet, otherwise they fall over, that is why in facultative-bipeds like kangaroo and some hadrosaurs have small front limbs and large tails it moves the center of gravity as close to the hips as possible. Basically anything the animal can do to move the center of gravity this way helps.
What exactly will help depends on what their anatomy and pelvic anatomy in particular is like overall. note that although the front limbs are skinny they are still close to the same length as the back limbs otherwise they cannot walk on all fours. If you look at hadrosaurs it looks like their front limbs are very short but they are the only dinosaur with free floating shoulders so the shoulder blades add to the limb length.
the limbs proportions also look different because the hind limb has to be able to support the entire weight of the animal while the front limbs only need to be able to support a a fraction of the weight. that is why dinosaurs also have larger pelvises compared to mammals, they need to be stronger and attack more muscle.
So, what you're asking for is called a facultative biped. There are many, many types of animals which are facultatively bipedal, even more than there are obligately bipedal animals. Some examples include pangolins, agamids, iguanas, sifakas, ornithopod dinosaurs, a lot of basal archosaurs like rauisuchians, prosauropods, ground sloths, chalicotheres, monitor lizards, beavers, raccoons, skunks, gibbons, indris, Eudibamus, baby sauropods, stegosaurs, tree kangaroos, and even cockroaches.
What's important here isn't the structure of the hip, but rather the distribution of mass. The main reason why the average quadruped can't walk on two legs is that too much of its mass needs its forelegs for support. If you have an animal where most of the weight is directly above the hips and hindlimbs, which never leave the ground, then you're good. Also, the hind legs should be longer than the forelegs.
There are lizard species that do this. For example Chlamydosaurus kingii, the Northern Australian Frilled Dragon that runs two-legged on land
and Basiliscus basiliscus, the Jesus Christ Lizard that can run on water!