How does mindfulness practice (sati) lead to jhana?

If you do 4 satipaṭṭhāna (equivalent to sati) with proper passaddhi (pacification/relaxation) awakening factor, then it's in the 4 jhāna spectrum of samādhi. If the passaddhi is insufficient, then it's just sati. If it's sufficient, it's both jhāna and sati. MN 125 is particularly clear in showing this, since it explicitly removes the first jhāna formula and replaces it with a second satipaṭṭhāna section. https://lucid24.org/mn/main/mn125/index.html#125.3.10.3 And here is a collection of all the sutta references I've collected over the years showing where sati and jhāna overlap, in all postures, activities, etc. https://lucid24.org/tped/s/samadhi/smd247/index.html MN 125 alone should clear up many modern misconceptions. If you can hear sounds, think verbal thoughts in sati, walk while doing sati, you can also do that in first jhāna. If you're in a samādhi where the mind is divorced from the body, you're no longer in 4 jhāna territory. You're in either in a formless attainment, or 8 vimokkha, or 8abhibhayata.

Is mindfulness the first jhana (Vitarka-vicāra)?

wikipedia vitakka-vicāra definition is not quite right. vitakka and vicāra is verbal, linguistic, communicable language. Otherwise, how could the Buddha's instruction, which is encoded in a communicable language, ever get decoded by your ears from sound, into a silent verbal mental talk of Dharma that sati and mind can think about? You hear dharma, memorize it (sati), and recollect and apply Dhamma (sati again). In first jhāna, this would be done with verbal thoughts (vitakka and vicāra). Beyond first jhāna, it's subverbal mental processing. For example, you cultivate an attitude of kindness and compassion, or your observe sensations of breath in the body, it doesn't require any verbal thought.
In short, sati can be verbal (with vitakka and vicāra), or subverbal.

The word mindfulness (sati) does not mean contemplating (anupassana).

Mindfulness refers to 'remembering' to keep the mind clear of unwholesome states.

When mindfulness keeps the mind clear of unwholesome states, the act of 'contemplating', i.e., 'seeing closely', naturally occurs.

When the mind is quiet & clear, the breathing & body naturally becomes the predominant meditation object. Then when the breathing calms down, feelings become the predominant meditation object. Then when the feelings calm down, the mind itself naturally becomes the predominant meditation object. Then when the mind becomes very clear & free, Dhammas or Realites naturally become the predominant meditation object.

When body, feelings, mind & Dhammas are 'seen closely', mindfulness again acts to ensure or 'remember' to abandon any wrong views that may arise towards body, feelings, mind & Dhammas.

Also, contemplating (anupassana) is a state of 'seeing', i.e., a state of 'non-active-thinking'. When the mind only sees & does not actively think, the mind calms down, the breathing calms down & the body calms down.

Thus, when when mindfulness maintains clarity & contemplating, certain 'nirodha' ('cessation') processes naturally, occur. The breathing becomes calmer & calmer, longer & longer, more & more refined. Internal stored stress progressively comes into awareness & dissolves. This calming & purification build a certain momentum, resulting in mindfulness being easier & easier to maintain because the mind, breath & body are becoming calmer & calmer; purer & purer. The happier & more content the mind is, the less effort required to maintain mindfulness.

When the body becomes very purified from stored mental stresses & formations, jhana will occur. This can be compared to continuously using mindfulness to rinse out dirt. When the dirt is finally rinsed out, water naturally flows into the cleared space. This naturally 'flowing' of the mind's momentum into a unity is jhana.

The mind flows into a 'unity' because the breathing has also become so refined that it cannot be felt anymore within the physical body. Therefore, instead of the mind naturally feeling/tracking the breathing in & out, down & up, within the body; the mind naturally creates a luminous mental image & absorbs into the mental image. This is jhana. The bliss that has generated within the body due to purification is now experienced intensely because the mind is absorbed into one-point with the mental image & the bliss is only felt in the brain.

In summary, mindfulness acts to remember, moment by moment, to keep the mind clear of unwholesome states & wrong views. This constant maintenance of mental clarity then, eventually, transforms into jhana.