The Wikipedia page on Pramana says:
Buddhism accepts only two pranama (tshad ma) as valid means to
knowledge: Pratyaksha (mngon sun tshad ma, perception) and Anumāṇa
(rjes dpag tshad ma, inference). Rinbochay adds that Buddhism also
considers scriptures as third valid pramana, such as from Buddha and
other "valid minds" and "valid persons". This third source of valid
knowledge is a form of perception and inference in Buddhist thought.
Valid scriptures, valid minds and valid persons are considered in
Buddhism as Avisamvadin (mi slu ba, incontrovertible, indisputable).
Perhaps, we can consider this third pramana the Śabda pramana, which according to this page is:
Sabda pramana is verbal testimony. It is also called ‘apta-vakyas’
(statement of a trust-worthy person’, and agama (authentic word).
In this essay entitled "Dhamma Without Rebirth?", Ven. Bhikkhu Bodhi wrote:
The aim of the Buddhist path is liberation from suffering, and the
Buddha makes it abundantly clear that the suffering from which
liberation is needed is the suffering of bondage to samsara, the round
of repeated birth and death. To be sure, the Dhamma does have an
aspect which is directly visible and personally verifiable. By
direct inspection of our own experience we can see that sorrow,
tension, fear and grief always arise from our greed, aversion and
ignorance, and thus can be eliminated with the removal of those
defilements. The importance of this directly visible side of Dhamma
practice cannot be underestimated, as it serves to confirm our
confidence in the liberating efficacy of the Buddhist path. However,
to downplay the doctrine of rebirth and explain the entire import of
the Dhamma as the amelioration of mental suffering through enhanced
self-awareness is to deprive the Dhamma of those wider perspectives
from which it derives its full breadth and profundity. By doing so one
seriously risks reducing it in the end to little more than a
sophisticated ancient system of humanistic psychotherapy.
Above, Ven. Bodhi states that part of the Buddha's teachings (Dhamma) is directly visible and personally verifiable, but he implies that the doctrine of rebirth is not. Therefore, this means that pratyaksha pramana is ruled out.
He also wrote in the same essay:
If we suspend our own predilections for the moment and instead go
directly to our sources, we come upon the indisputable fact that the
Buddha himself taught rebirth and taught it as a basic tenet of his
teaching.
In the Maha-Saccaka Sutta (MN36), we have rebirth supported using the Śabda pramana, based on the Buddha's testimony to Saccaka, the Jain:
"When the mind was thus concentrated, purified, bright, unblemished,
rid of defilement, pliant, malleable, steady, & attained to
imperturbability, I directed it to the knowledge of recollecting my
past lives. I recollected my manifold past lives, i.e., one birth,
two...five, ten...fifty, a hundred, a thousand, a hundred thousand,
many eons of cosmic contraction, many eons of cosmic expansion, many
eons of cosmic contraction & expansion: 'There I had such a name,
belonged to such a clan, had such an appearance. Such was my food,
such my experience of pleasure & pain, such the end of my life.
Passing away from that state, I re-arose there. There too I had such a
name, belonged to such a clan, had such an appearance. Such was my
food, such my experience of pleasure & pain, such the end of my life.
Passing away from that state, I re-arose here.' Thus I remembered my
manifold past lives in their modes & details.
"This was the first knowledge I attained in the first watch of the
night. Ignorance was destroyed; knowledge arose; darkness was
destroyed; light arose — as happens in one who is heedful, ardent, &
resolute. But the pleasant feeling that arose in this way did not
invade my mind or remain.
"When the mind was thus concentrated, purified, bright, unblemished,
rid of defilement, pliant, malleable, steady, & attained to
imperturbability, I directed it to the knowledge of the passing away &
reappearance of beings. I saw — by means of the divine eye, purified &
surpassing the human — beings passing away & re-appearing, and I
discerned how they are inferior & superior, beautiful & ugly,
fortunate & unfortunate in accordance with their kamma: 'These beings
— who were endowed with bad conduct of body, speech, & mind, who
reviled the noble ones, held wrong views and undertook actions under
the influence of wrong views — with the break-up of the body, after
death, have re-appeared in the plane of deprivation, the bad
destination, the lower realms, in hell. But these beings — who were
endowed with good conduct of body, speech & mind, who did not revile
the noble ones, who held right views and undertook actions under the
influence of right views — with the break-up of the body, after death,
have re-appeared in the good destinations, in the heavenly world.'
Thus — by means of the divine eye, purified & surpassing the human — I
saw beings passing away & re-appearing, and I discerned how they are
inferior & superior, beautiful & ugly, fortunate & unfortunate in
accordance with their kamma.
The following arguments come from the book "Introduction to Abhidhamma" by Nina van Gorkom, from this chapter. This is likely to be the author's personal opinion that karma and rebirth can be inferred from the fact that sentient beings are born with very different tendencies and inclinations:
In order to understand what causes birth we should know what
conditions the nāma and rūpa which arise at the first moment of a new
lifespan. The citta which arises at that moment is called the
rebirth-consciousness or paṭisandhi-citta. Paṭisandhi means relinking,
it “ links” the previous life to the present life. It is usually
translated as rebirth-consciousness, but, since there is no person who
is reborn, birth-consciousness would be more correct. Since there
isn’t any citta which arises without conditions, the paṭisandhi-citta
must also have conditions. The paṭisandhi-citta is the first citta of
a new life and thus its cause can only be in the past. One may have
doubts about past lives, but how can people be so different if there
were no past lives? We can see that people are born with different
tendencies and talents. Cittas which arise and fall away succeed one
another and thus each citta conditions the next one. The last citta of
the previous life (dying-consciousness) is immediately succeeded by
the first citta of this life, without there being any interval. That
is why tendencies one had in the past can continue by way of
accumulation from one citta to the next one and from past lives to the
present life. Since people accumulated different tendencies in past
lives, they are born with different tendencies and inclinations.
Rebirth-consciousness is the result of kamma, it is vipākacitta. Our
life starts at the moment the paṭisandhi-citta arises together with
the rūpa which is at the same time produced by kamma. A lifespan ends
when the last citta, the dying-consciousness (cuti-citta) falls away.
Kamma produces rūpa not only at the first moment of life but
throughout our life.
Apart from the Abhidhamma, it is likely that arguments could be found from Nagarjuna's Mūlamadhyamakakārikā or other Indian Mahayana writings, but I'm not sufficiently familiar with them.
You may also find Ven. Thanissaro's essay "The Truth of Rebirth And Why it Matters for Buddhist Practice" useful. He wrote:
To avoid wrong view — and the ridicule it deserves — the Buddha found
it necessary to disclose his knowledge that there are lives after
death. And he had to include the perspective not just of one lifetime
after death, but of many. ....
This is why the Buddha never claimed to offer proof for either the
efficacy of action or for rebirth, for he knew that the evidence for
these teachings lay beyond the ken of most of his listeners. ....
These arguments don't prove the efficacy of action or the truth of
rebirth, but they do show that it is a safer, more reasonable, and
more honorable policy to assume the truth of these teachings than it
would be to assume otherwise. The Buddha didn't press these arguments
beyond that point.
Also comparing the Buddhist view to the Hindu and Jain views, he wrote:
In discussing rebirth, the Buddha differed from the other schools of
the time in that he didn't base his position on a metaphysical view of
personal identity — that is, on defining what it is that gets reborn.
By placing rebirth in the context of dependent co-arising, he was
presenting it in a phenomenological context — i.e., one that focused
on phenomena as they can be directly experienced and that refused to
take a stand on whether there is a reality of "things" underlying
them.