Father-of-two Sergeant Mark Foley, 31, who served in Afghanistan, was killed in Warcop, Cumbria, while travelling in a military vehicle.
An investigation into the accident, which injured two other soldiers, is under way.
Paying tribute to her husband, who served with Norfolk-based The Light Dragoons but was originally from Hull, Kelly Foley said: “He was my soulmate and my best friend, who always made me laugh. The best dad ever to Emily and Hannah. He loved us, his job and Hull FC. We will miss him so much.”
His commanding officer, Lieutenant Colonel James Senior, said: “Sergeant Foley was an exceptionally talented, battle-hardened soldier who embodied everything that is good about our profession.
“A consummate professional, he delivered on operations time and time again, accumulating huge experience on the way.
“Always inquisitive and engaging, Sgt Foley was at the heart of everything his troop did, everything his squadron did, everything the Regiment did.
“Whether in the dust of Helmand Province or the troop store at our Regiment’s home in Norfolk, he was the man people turned to for advice and guidance.
“Sgt Foley made an astonishing contribution to our Regimental family, and his loss is a devastating shock.
“Our thoughts and prayers are with his wife and young family.”
An Army spokeswoman said a specialist team was investigating the cause of the accident at the training centre.