Corrie Mckeague, from Dunfermline, has been missing since going on a night out in Bury St Edmunds, Suffolk, on 24 September.
His mother Nicola Urquhart said she had "no faith" the police would find him.
On a JustGiving page set up on Tuesday night, 1,788 people have pledged £20,336 to the fund.
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The JustGiving page was set up by Cheryl Hickman, who runs The Bull Inn at Barton Mills where Mrs Urquhart has been staying since her son went missing.
The funding bid for a private investigator was set up after Mrs Urquhart questioned Suffolk Police resources.
She said the force had "behaved in a way" which had "absolutely destroyed" her belief that they are "competent and they know what they're doing".
Temporary Assistant Chief Constable Louisa Pepper said she has confidence in her team of officers.
She said the force was using an "inordinate" amount of resources in the hunt for Mr Mckeague.
He was last seen on CCTV walking alone and eating fast food in Brentgovel Street, Bury St Edmunds, at about 03:20 BST.
A five-figure reward for information on his whereabouts has been offered by his grandparents.
A crowdfunding campaign to hire a private investigator to search for a missing serviceman has topped its £20,000 target in less than two days.