How is craving the practice that leads to the origin of feeling?
The Mahayana answer would be that Vedana does not refer to feeling arising from an actual contact with an actual external object. Instead, Vedana is a subjective experience of feeling attributed to a subjective experience of contact with a subjective idea of an object. Basically, Vedana is a thought: "that thing that was over there, now that I have it right here, gives me this feeling of comfort (or this feeling of discomfort, or this neutral feeling)".

So, while Vedana depends on a subjective experience of contact as its immediate cause, more fundamentaly it depends on that idea of a desirable external object.

According to Mahayana interpretation of Twelve Nidanas, the process of dependent origination is actually cyclical, not linear. It's explained in twelve steps for ease of presentation but in actuality it represents a spiral of a self-enforcing tendency. The elements that later become Craving and Vedana are present in rudimentary/latent form even at the Formations(Tendencies) stage (the second Nidana) and get stronger as things get more concrete. 

So at the root of the process is ignorant tendency of the basic mind to get settled into certain experiences which are then considered comfortable, and their opposites are considered uncomfortable. As the mind develops it learns to seek those comfortable experiences and avoid uncomfortable ones. This tendency to seek and avoid develops into ability to delineate entities (objects and the subject) and attribute qualities to them based on one's attitude. The experience of those qualities at the time of subjective "contact" is Vedana. 

So Vedana, both comfortable (but transient, and therefore eventually uncomfortable) and directly uncomfortable, depends on this tendency to seek comfortable experiences attributed to external entities. That's what Craving refers to in short. After all, craving is always craving for an external something because of its implied promise to bring an internal experience of the right nature.

That's why Buddha says Craving is a practice (or path) that leads to Vedana. Because we attribute qualities arising in our mind to external entities, then we crave and pursue those entities (or avoid them), then we experience something as result, then we identify that experience as coming from the entity. This is Vedana.

Attributing perceptual qualities that in fact come from our mind to entities we have delineated out of totality of our experience, and then forming attitude to those entities, of either craving or avoidance, is the activity that leads to the illusory experience of pain and happiness.