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An answer to a similar question discusses limitations of House Elves. If House Elves couldn't be used to search Chamber of Secrets, why didn't he simply asked ghosts?
Ghosts can certainly move around freely without any bound.
If Moaning Myrtle was shy to be interacted, he could order house ghosts like Nearly Headless Nick or Bloody Baron. Also, Professor Binns could have happily accepted his request.
Why didn't Dumbledore ask ghosts to search for the Chamber of Secrets?
We don't know, but maybe it was done in the past
Remember that the Basilisk can turn ghosts into stone. Perhaps, in the past, people did ask ghosts to help search for the Chamber of Secrets. If those searching ghosts started going missing because they found the basilisk, other ghosts would likely be reluctant to look as well.
In this writing by JKR in Pottermore:
Having chosen a feeble simulacrum of mortal life, ghosts are limited in what they can experience. No physical pleasure remains to them, and their knowledge and outlook remains at the level it had attained during life, so that old resentments (for instance, at having an incompletely severed neck) continue to rankle after several centuries.
We may assume that they can't go where they want because they are just an imprint of what they were. So I think they can only haunt what they knew in live.
We have Myrtle as an example to leave the place she died to haunt Olive Hornby but maybe ( not sure ) it's because she haunted Olive and not the places she was. She knew Olive.
