TL;DR: Udagawa could be right, but he does not present enough concrete evidence to overturn the prevailing theory or exclude Portugese involvement. What is important is that the gun was introduced via a trade network in which the wakō "pirates" were involved, so the wakō very much played a part in the introduction of the gun into Japan.

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That you even heard about Udagawa's work means I can't really offer anything in the way of confirmation as it is at the forefront of research and debate. I can, however, offer my stance on the topic.

Udagawa relies on two evidence for his theory, as there are no written account to back him up (sort of, I'll get into it). First, if the arquebus was introduced solely from the Tanegashima incident, then we should expect to find earliest Japanese guns to be of the same design, all copied from the Tanegashima gun. But we don't. Second, if Japanese guns were introduced by Europeans, we should expect Japanese guns and parts to look like European guns, but they don't (outside of the general long-barrel firearm configuration). Instead, they look more like guns found in Southeast Asia at the time.

The first fact I can not argue either way. I wish I could examine the weapons myself. If true, it could be evidence for multiple introductions of the firearm at around the same time. Or it could be evidence that there were many different styles of firearms employed in the area at the time. Either way, it wouldn't be evidence against Portuguese involvement. Especially as we have written records of Portuguese contact in the time frame 1542-1544 in Portuguese sources, and of course the Teppōki and the Tanegashima kafu used for the traditional theory. And none (or at least, none found yet), that the guns were introduced from types bought in Southeast Asia, and of course the fact that guns were called Tanegashima.

The second is true. But it was already known that the Japanese Tanegashima is different from the European infantry arquebus/musket. But Udagawa doesn't present enough evideince to say why. Europe had many different types and configurations for firearms as well. Just because the Tanegashima had a different configuration than "standard" European infantry arquebus does not mean its origins was not European. For instance the top gun in the picture here much better resembles the Tanegashima, but is very much European, and earlier than the Tanegashima. And of course, even if the Tanegashima was indeed copied off Southeast Asian style, the question becomes whether that style originated in Europe or Southeast Asia, as Europeans already have enough of a hold to influence firearms development in the area. So once again, what Udagawa presented as evidence can not be used to rule out possible, or even probable, Portuguese involvement.

So where does that leave us? I think what Udagawa reminds us is this: we need to remember the Japanese side of the introduction story, and the context of that introduction.

At the time, there was a vibrant trade network between China, Ryūkyū, Kyūshū, and Southeast Asia. It was through this network that the gun was introduced into Japan. The Portuguese was involved in this network, but so was the Japanese. Clans who lived on the coast, especially those in Kyūshū and other islands to the southwest were heavily involved in foreign seaborne trade and piracy. And of course, the Chinese were heavily involved as well. Wakō, by ethnicity, was predonminantly Chinese after all. Also don't forget everyone else in the area. The only difference between a pirate ship, a clan's warship, and a merchant ship was what the ship and crew was authorized to do and was actually doing. There's no reason to rule out the Tanegashima clan and islanders from being involved in the same activity. And of course according to the traditional theory via the teppōki, the ship that brought the two Portugese was a Chinese junk and the one who translated for them was a (Chinese) wakō pirate/merchant boss (might have been his ship) who had good contacts with the clans in the southwestern islands. Even ignoring the rest of Udagawa's theory, this alone demonstrates that the wakō was very much involved in the introduction of the gun into Japan.