Mr Dunn, 19, was killed when his motorbike crashed into a car allegedly being driven by American Anne Sacoolas outside US military base RAF Croughton in Northamptonshire on August 27 last year.
Diplomatic immunity was asserted by the US on behalf of Sacoolas following the collision and she was able to return to her home country, sparking an international controversy.
Donald Trump later called the incident a “terrible accident”.
President Trump told reporters: “When you get used to driving on our system and then you’re all of a sudden on the other system where you’re driving – it happens. You have to be careful, very careful.”
In a video message congratulating Mr Biden, Mr Dunn’s mother Charlotte Charles said: “It’s wonderful news to us and I wish him and his team much success over the coming years.
“It’s given myself and team Harry some renewed hope of getting justice for our boy.
“He still fully deserves justice, as do we.
“We hope that seeing that Mr Biden, through personal experience, can have a far deeper understanding of our pain and our suffering since we lost Harry.”
She pleaded for Mr Biden to “send Mrs Sacoolas back”.
Mr Biden has often spoken of the loss of his first wife Neilia and baby daughter Naomi in a car accident in 1972, which also left his toddler sons Beau and Hunter injured.
Last month, Ms Charles reached out to Mr Biden in a video message in which she said her “heart goes out” to him after Beau died of a brain tumour, aged 46, in 2015.