Peter Nally and the three school-aged children died when the five-seater Cirrus they were travelling in crashed then caught alight in a paddock at Gundaroo, north of Queanbeyan, on Friday afternoon.
Local media reports said Mr Nally was a veteran pilot and his grandchildren were aged 11, nine and six.
Their family said on Sunday there were "no words" to describe their grief.
"As we are still coming to terms with our loss, we ask that the media respect our privacy while we grieve and process this tragedy," they said in a statement shared by NSW Police.
"We are appreciative of the overwhelming support we continue to receive from our extended family and friends as well as the local community."
The Australian Transport Safety Bureau (ATSB) is investigating the cause of the crash.
The Redcliffe Aero Club expressed its condolences to those killed and said the pilot was "active in the social side of the club with many hours of flying experience".
Flight information shows the single-engine aircraft arrived in Canberra on Wednesday after travelling via Armidale from Redcliffe, north of Brisbane.
It's understood the plane vanished from radar screens 16 minutes after taking off on Friday.
ATSB crews will monitor for spilled chemicals, burnt carbon fibre and possible explosive materials.
"This will be a difficult undertaking given how burnt the wreckage is," the bureau's Colin McNamara said.
"But over the coming days, I am fully confident that the investigators will gather as much information and evidence as they can from the accident site."
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