How is craving the practice that leads to the origin of feeling? SN 36.23 translated by Bhikkhu Sujato says: 


  Then a mendicant went up to the Buddha, bowed, sat down to one side,
  and said to him:
  
  Atha kho aññataro bhikkhu yena bhagavā tenupasaṅkami; upasaṅkamitvā
  bhagavantaṃ abhivādetvā ekamantaṃ nisīdi. Ekamantaṃ nisinno kho so
  bhikkhu bhagavantaṃ etadavoca:
  
  “Sir, what is feeling? What’s the origin of feeling? What’s the
  practice that leads to the origin of feeling?
  
  “katamā nu kho, bhante, vedanā, katamo vedanāsamudayo, katamā
  vedanāsamudayagāminī paṭipadā?
  
  What’s the cessation of feeling? What’s the practice that leads to the
  cessation of feeling?
  
  Katamo vedanānirodho, katamā vedanānirodhagāminī paṭipadā?
  
  And what is feeling’s gratification, drawback, and escape?”
  
  Ko vedanāya assādo, ko ādīnavo, kiṃ nissaraṇan”ti?
  
  “Mendicant, there are these three feelings:
  
  “Tisso imā, bhikkhu, vedanā—
  
  pleasant, painful, and neutral.
  
  sukhā vedanā, dukkhā vedanā, adukkhamasukhā vedanā.
  
  These are called feeling.
  
  Imā vuccanti, bhikkhu, vedanā.
  
  Feeling originates from contact.
  
  Phassasamudayā vedanāsamudayo.
  
  Craving is the practice that leads to the origin of feeling.
  
  Taṇhā vedanāsamudayagāminī paṭipadā.
  
  When contact ceases, feeling ceases.
  
  Phassanirodhā vedanānirodho.
  
  The practice that leads to the cessation of feelings is simply this
  noble eightfold path, that is:
  
  Ayameva ariyo aṭṭhaṅgiko maggo vedanānirodhagāminī paṭipadā,
  seyyathidaṃ—
  
  right view, right thought, right speech, right action, right
  livelihood, right effort, right mindfulness, and right immersion.
  
  sammādiṭṭhi … pe … sammāsamādhi.
  
  The pleasure and happiness that arise from feeling: this is its
  gratification.
  
  Yaṃ vedanaṃ paṭicca uppajjati sukhaṃ somanassaṃ, ayaṃ vedanāya assādo;
  
  That feeling is impermanent, suffering, and perishable: this is its
  drawback.
  
  yaṃ vedanā aniccā dukkhā vipariṇāmadhammā, ayaṃ vedanāya ādīnavo;
  
  Removing and giving up desire and greed for feeling: this is its
  escape.”
  
  yo vedanāya chandarāgavinayo chandarāgappahānaṃ, idaṃ vedanāya
  nissaraṇan”ti.


Bhikkhu Bodhi's translation is more accurate, here:


  There are, bhikkhu, these three feelings: pleasant feeling, painful
  feeling, neither-painful-nor-pleasant feeling. This is called feeling.
  With the arising of contact there is the arising of feeling. Craving
  is the way leading to the origination of feeling. With the cessation
  of contact there is the cessation of feeling. This Noble Eightfold
  Path is the way leading to the cessation of feeling; that is, right
  view … right concentration.


In the above sutta and often elsewhere, it is said" "With the arising of contact there is the arising of feeling". Also, other suttas say: "With the arising of feeling there is the arising of craving". 

This being so, 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.