The vehicle had been on its way to a military site in Al-Bayda when the charge went off, overturning the vehicle and injuring the soldiers, residents told the Anadolu Agency on Wednesday.
The explosion had taken place at a mountainous road near Al-Bayda city, the residents said.
No group has yet claimed responsibility for the attack, on which the Yemeni army has also yet to comment.
However, the government typically blames Al-Qaeda-affiliated Ansar al-Sharia militant group for similar, mostly deadly, attacks on military personnel in several parts of the country.
Ansar al-Sharia movement said in a statement late on Saturday that it had carried out 205 attacks in 11 Yemeni provinces during the past three months.
The movement added that most of these attacks had targeted members of the Shiite Houthi movement.
The Houthis seized control of Yemeni capital Sanaa in September before moving on to establish control over other parts of the country as well.
The rise of the Houthis has pitted the Shiite group against local Sunni tribes and Al-Qaeda, the latter of which is said to remain active in Yemen.
Yemen has remained in the throes of turmoil since President Ali Abdullah Saleh stepped down in 2012 under pressure from a months-long popular uprising against his 33-year rule.