According to Buddha's characteristics and teaching in Pali Canon, could he have taught Vajrayana (and Karmamudra)?

While Buddha did say "I have taught the Dhamma, Ananda, making no inner and outer", he was known to adapt his message to the audience. It is rather apparent from the suttas that Buddha did not teach lay people his more advanced doctrines e.g. Pratityasamutpada.

Similarly in Vajrayana, the elements of Dharma are introduced in a progression, not skipping to advanced teaching until student gets a firm grasp of the basics. Just as Buddha rightly considered Pratityasamutpada too advanced for untrained lay people, in Vajrayana advanced teaching such as Madhyamaka, Lower Tantra, Mahamudra, and Dzogchen is rightly considered too much for most beginners. In fact, advanced levels of Vajrayana (esp. Dzogchen) are often characterized as "self-secret" because they are simply incomprehensible to an unprepared student.

The way Vajrayana sees itself (if I can be as arrogant as to speak on its behalf) is as a natural elaboration of principles implicit in Buddha's teaching, similarly to how Theravada's Visuddhimagga introduces hundreds of concepts Buddha never spoke about in suttas. Think modern mathematics, which can be seen as development of principles laid down by Pythagoras, or modern western scientific method as an extension of Aristotle's work. The fact that Pythagoras did not teach calculus does not mean it's not real mathematics.

As I said in How do taboo acts work in tantra if people don't see the acts as taboo?, Kamamudra is a very small part of Vajrayana and much of it is about solving emotional issues that are unique to the current society dominated by abstract concepts rather than by natural relationships as was back in Buddha's times.

The Buddha didn't teach karmamudra. Read the Pali canon, in which the whole doctrine is laid out.

No, the canon doesn't approve of that at all.

Then Ven. Ananda approached the nun and, on arrival, sat down on a prepared seat. As he was sitting there, he said to the nun: "This body, sister, comes into being through food. And yet it is by relying on food that food is to be abandoned.

"This body comes into being through craving. And yet it is by relying on craving that craving is to be abandoned.

"This body comes into being through conceit. And yet it is by relying on conceit that conceit is to be abandoned.

"This body comes into being through sexual intercourse. Sexual intercourse is to be abandoned. With regard to sexual intercourse, the Buddha declares the cutting off of the bridge. -- AN 4.159

According to the early texts the Buddha didn't have any secret teachings. Somewhere in the Digha Nikaya (I think it's in the Mahaparinibbana Sutta, but it's a very long Sutta and I can't quite locate it) the Buddha said: "I do not have the closed fist of the teacher who holds anything back" and in AN 1.283 the Buddha said: "Three things shine openly, not in secret. What three? The orb of the moon, the orb of the sun and the Dhamma and discipline taught by the Tathàgata"

One thing to note though is that the Tantras mostly don't claim to have been spoken by the historical Buddha Sidhartha Gautama, or Shakyamuni Buddha, but by Vajradhara, a different Buddha that is often understood to be Shakyamuni appearing in Tantric form. They say that Vajradhara appeared to the various Mahasiddhas in visions and revealed to them the various Tantras that became the basis of the tantric teachings.

Also, just to clarify for readers who may not be aware, Karmamudra is a high level tantric practice in which one engages in sexual intercourse with a special consort in order to destroy the knots around the Heart Chakra and thus allow the attainment of enlightenment. It is a very high level practice and isn't done lightly. Monks are not allowed to do it (at least in the Gelugpa school) because of their vows of celibacy. It's a very rare practice.