Smith has been bailed to appear before East Yorkshire Magistrates Court later this month.
Locke and outside-back Lyne are not facing police action, but were reportedly with Smith at the time, according to broadcaster Dan Ginnane.
Speaking on Triple M radio in Sydney, Ginnane said he received reports of the trio stealing the car from their coach.
His sources told him the car the trio were driving was allegedly stolen from Brian Smith, before they crashed it into a wall.
"Tim Smith, who plays for Wakefield along with Kevin Locke, got on the drink, stole their coach Brian Smith's car and put it into a wall - this was the story I was told," said Ginnane.
"Even for rugby league that's ridiculous.
"We thought, 'It's clearly a furphy, it's even too outrageous for rugby league'. But since then it has emerged Tim Smith has been arrested for DUI. and for an incident - presumably an accident.
"Can you imagine if this is true, that two or three players have stolen the coach's car and put it into a wall?"
Wakefield released a statement saying: "A full investigation has been launched and the club are co-operating fully with Humberside Police, the RFL and RL Cares in this matter".
The reports have not been confirmed at this stage.
