On p. 8 Tintin becomes aware that a man was kidnapped (inspector Bunji Kuraki). Immediately after that, he received a phone call from the sagacious Thom(p)son.
Actually the Thompsons call Tintin the following morning. That is what is stated in the text box above the ringing phone and Tintin is shown in pajamas and dressing-gown when talking to them.
When the Thompsons call Tintin it is only after they have identified the drowned man as Herbert Dawes. He had no ID on him when he was fished out of the river so identifying him would have taken some time - like a friend worrying about him, going to the police, identifying him in the morgue and then giving details such as the fact that he served on the "Karaboudjan".
I imagine that after the kidnapping, the regular police would have been warned and questioned Tintin about it. He may not have necessarily have mentioned the word "Karaboudjan" which, when he found it on the piece of paper, meant nothing to him.
Even if he had, the police may not necessarily have followed it up. Who would have connected the name "Karaboudjan" to a cargo ship? We are talking about the 1940s when records were in large books or registers and going through them would have taken days - not seconds like on a computer today.
In any case, Tintin is the sort who likes to handle cases himself. Take for instance the "Broken Ear": when the "fetish" is returned to the museum, Tintin does not tell the Thompsons that it is a fake; and he does not appear to warn them that Perez and Bada have burgled his flat (for the parrot) and tried to kill him. He prefers to handle it himself and arrests them when he has them bang to rights for the actual murder of Tortilla.
Also, Tintin has no idea who the kidnapped man is or if there is a connection to Dawes' drowning. When the Thompsons go to the "Karaboudjan" and talk to the mate, Alan, they are quite satisfied with his story that Dawes would have drowned as a result of being drunk. They have no reason to connect him to the kidnap and thus no reason to search the ship.