TL;DR: Just because House-Elves can Apparate to places where wizards cannot, it doesn't mean that they can Apparate to every destination.
The exact location of the Chamber of Secrets was unknown and unseen for centuries. No one, except from the Heir of Slytherin, has ever visited this place and as far as I am concerned, there was no known indication in in-universe written material describing the interior of the place.
Although House-Elf Apparition is not bound to the same restrictions as the wizards' counterpart, the elf might still need to visualize the target location. JK Rowling has already confirmed that this the case for wizards:
Apparition requires knowledge of the terrain to which one is moving, or the ability to visualise it clearly.
JK Rowling FAQ, 20 December 2016
Keep in mind that Kreacher, who was able to Apparate to the cave with Regulus Black, has first visited the location with Voldemort himself. Kreacher didn't just "imagined" where they would like to go, with the elf magic doing the rest.
By taking this into account, Dumbledore couldn't just ask the House-Elves to try to Apparate there, even if he had managed to stand in front of the Chamber's gate. No House-Elf would have seen what's inside the gate, thus preventing them to Apparate. If that wasn't the case, Dumbledore would have used a House-Elf to enter the cave that the Locket-Horcurx was hidden and wouldn't travelled there or spent so much time searching it across the country, gathering memories of people close to Voldemort.
Moreover, the Chamber of Secrets was probably being protected with some sort of Anti-Apparition charm that was preventing the Apparition of House-Elves too, apart from wizards.
Note regarding Fawkes's Apparition
In Harry Potter and the Chamber of Secrets we see that Fawkes is able to Apparate into the Chamber, without the prior knowledge of the interior of the place. It is also stated that Fawkes did not fly all over from the Headmaster's office through the hole that Ron had made, as he was surprised seeing the phoenix there:
"Where'd that bird come from?!"
Chapter 17, "The Heir of Slytherin", Harry Potter and the Chamber of Secrets
Yet again, the magic of Fawkes might be different than the magic of House-Elves. Moreover, don't forget the fact that according to Dumbledore, Fawkes was summoned there due to his loyalty to Harry. That implies that Fawkes Apparated straight to where Harry was being located at that time, regardless of the lack of knowledge of the interior of the Chamber, the exact location of it or possible prevention charms that were put in place.
Phoenixes are probably able to find specific people without knowing the location of them, as Owls can find specific people to deliver letters, even if they were never given their address. The difference is that the former ones use Apparition to get there while the latter ones travel to them by flying.
(The assumption I've made previously about Dumbledore not using a House-Elf to directly Apparate into the cave of the Horcrux without knowing the location of it, applies here also.)
Note regarding Dobby's Apparition to Shell Cottage
Others have suggested, that Dobby was able to Apparate to Shell Cottage without ever seeing it. That's totally not true. First, Shell Cottage was being shielded by the Fidelius Charm thus no one, not even a House-Elf, could have entered the Charm's field if the Secret-Keeper hasn't revealed them the secret.
Secondly, Dobby Apparated somewhere near Shell Cottage, following the instructions Ron explicitly gave to him (emphasis mine):
"Right. Dobby, I want you to grab Luna, Dean and Mr Ollivander, and take them ‒ take them to ‒"
"Bill and Fleur's", said Ron. "Shell Cottage on the outskirts of Tinworth!"
The elf nodded for a third time.
Chapter 23, "Malfoy Manor", Harry Potter and the Deathly Hallows
Ron provided something like an "address" to Dobby which made them Apparate near the house of Bill and Fleur and not inside it.