These two ballots are the first to be held under the agreement reached between GMB and the company in June. The deal provides the mechanism whereby employees can vote for trade union recognition and collective bargaining if they wished to do so.
The GMB now has collective bargaining rights in 11 of the supermarket's 24 depots.
Paul Kenny, GMB general secretary, said: "This exercise has been a great success for GMB, for Asda and most of all for the employees at Chepstow and Erith depots. It is important to know how strongly Asda workers value having an independent trade union voice at work."
Asda said in a statement: "We have always been committed to listening to all our colleagues on issues that affect them and that's exactly what these ballots were for – to ask all our colleagues at our Chepstow and Erith depots, union and non-union members alike, whether they wanted to move to having a collective agreement with GMB at their site."