How, if at all, does karma exist in the form and formless realms?

The Buddha never called the "human realm" the "desire realm". The Buddha described the "human realm" as follows:

SN 56.61 The beings born (paccājāyanti) as humans (manussesu) are few, while those not born as humans are many. Why is that? It is because they have not seen the four noble truths.

AN 6.39 Bhikkhus, a god, a human or any other good state would not be evident from actions born of greed, hate and delusion. Yet, bhikkhus, from actions born of greed, hate and delusion a hellish being, an animal birth a ghostly birth or some other bad state would be evident.

SN 56.47 Sooner, I say, would that blind turtle, coming to the surface once every hundred years, insert its neck into that yoke with a single hole than the fool who has gone once to the nether world would regain the human state. For what reason? Because here, bhikkhus, there is no conduct guided by the Dhamma, no righteous conduct, no wholesome activity, no meritorious activity. Here there prevails mutual devouring, the devouring of the weak. For what reason? Because, bhikkhus, they have not seen the Four Noble Truths. What four? The noble truth of suffering … the noble truth of the way leading to the cessation of suffering.

Kamma obviously can exist in the form and formless realms, because these realms can be a type of becoming caused by craving, as follows:

SN 12.2 And what is becoming? These three are becomings: sensual becoming, form becoming, & formless becoming. This is called becoming.

AN 3.76 "Ananda, if there were no kamma ripening in the sensuality-property, would sensuality-becoming be discerned?”

“No, lord.”

“If there were no kamma ripening in the form-property, would form-becoming be discerned?”

“No, lord.”

“If there were no kamma ripening in the formless-property, would formless-becoming be discerned?”

“No, lord.”

Naturally, the bliss of the form & formless realms is impermanent, as follows:

AN 4.123 Again, there is the case where an individual, with the stilling of directed thoughts & evaluations, enters & remains in the second jhana: rapture & pleasure born of composure, unification of awareness free from directed thought & evaluation — internal assurance. He savors that, longs for that, finds satisfaction through that. Staying there — fixed on that, dwelling there often, not falling away from that — then when he dies he reappears in conjunction with the Abhassara[1] devas.[2] The Abhassara devas, monks, have a life-span of two eons. A run-of-the-mill person having stayed there, having used up all the life-span of those devas, goes to hell, to the animal womb, to the state of the hungry shades. But a disciple of the Blessed One, having stayed there, having used up all the life-span of those devas, is unbound right in that state of being. This, monks, is the difference, this the distinction, this the distinguishing factor, between an educated disciple of the noble ones and an uneducated run-of-the-mill person, when there is a destination, a reappearing.

MN 111 Furthermore, with the complete transcending of the dimension of the infinitude of consciousness, [perceiving,] 'There is nothing,' Sariputta entered & remained in the dimension of nothingness. Whatever qualities there are in the dimension of nothingness — the perception of the dimension of nothingness, singleness of mind, contact, feeling, perception, intention, consciousness, desire, decision, persistence, mindfulness, equanimity, & attention — he ferreted them out one after another. Known to him they arose, known to him they remained, known to him they subsided. He discerned, 'So this is how these qualities, not having been, come into play. Having been, they vanish.' He remained unattracted & unrepelled with regard to those qualities, independent, detached, released, dissociated, with an awareness rid of barriers. He discerned that 'There is a further escape,' and pursuing it there really was for him.

Taking a Theravada view, the immaterial realms - what you call formless - occur when a practitioner dies whilst dwelling in one of the four arupa ayatanas:

1. The dimension of the perception of space
2. The dimensions of the perception boundless consciousness
3. The dimensions of the perception of nothingness
4. The dimension of neither perception nor non-perception

Because these states still contain figments of conditionality, after death the mind (citta) gravitates to those figments thus perpetuating the continuation of that respective consciousness. As I currently understand, the dimension of neither perception nor non-perception is considered to be at the very prelude to the finality of reality in the grand scheme of dhamma: the figments of consciousness that exist there are too tenuous to cause any major form-based re-becoming disturbances. The task here is traversing the pleasure of being relinquished from those form-based consciousnesses.

I have a long held a contention about the commonly held view in Theravada concerning rupa jhana and arupa ayatanas. The contention concerns the progressive models from jhana to ayatana that are often touted. In my view they are very distinct and either set doesn't necessarily inform the other set.

Back to the question: what we could say is that the practitioner, working in the sensuous domain, having previously developed his mind in accordance with dhamma, and having passed away, now finds himself dwelling in some heightened pleasurable state; one of the four arupa ayatanas. This could be said to have happened because the cause of that final state was due to the practice: due to this karma the practitioner traversed from the form state to the formless.