Ser Vardis Egen was the captain of Jon Arryn’s household guard, an honorable position. Multiple reasons forced Ser Vardis Egen to fight or die at the hands of Bronn, some of these are:
Honor of representing the House Arryn:
Ser Vardis was selected over other knights who wanted to fight Tyrion Lannister.
Lady Lysa raised a hand for silence. “I thank you, my lords, as I know my son would thank you if he were among us. No men in the Seven Kingdoms are as bold and true as the knights of the Vale. Would that I could grant you all this honor. Yet I can choose only one.” She gestured. “Ser Vardis Egen, you were ever my lord husband’s good right hand. You shall be our champion.”
A Game of Thrones, Chapter 40, Cateyln
Bronn was a sellsword:
Sellswords are not respected by Knights. It would have been disrespectful/insulting for Ser Vardis to surrender to a sellsword.
Ser Morton Waynwood said. “Ser Vardis is a knight, sweet lady. This other fellow, well, his sort are all cowards at heart. Useful enough in a battle, with thousands of their fellows around them, but stand them up alone and the manhood leaks right out of them.”
A Game of Thrones, Chapter 40, Cateyln
Ser Vardis was fighting with Jon Arryn's sword:
Lysa had given Jon Arryn's sword to Ser Vardis. It was considered an honor to the old knight.
Ser Vardis held out a gauntleted hand, and his squire placed a handsome double-edged longsword in his grasp. The blade was engraved with a delicate silver tracery of a mountain sky; its pommel was a falcon’s head, its crossguard fashioned into the shape of wings. “I had that sword crafted for Jon in King’s Landing,”
"I thought it only fitting that our champion avenge Jon with his own blade."
A Game of Thrones, Chapter 40, Cateyln
Loyalty to Lady Lysa:
Even when most knights realized Ser Vardis was in trouble, Lady Lysa commanded him to fight on and he fought.
Blind with arrogance as they were, even the knights and lords of the Vale could see what was happening below them, yet her sister could not. “Enough, Ser Vardis!” Lady Lysa called down. “Finish him now, my baby is growing tired.”
And it must be said of Ser Vardis Egen that he was true to his lady’s command, even to the last. One moment he was reeling backward, half-crouched behind his scarred shield; the next he charged. The sudden bull rush caught Bronn off balance.
A Game of Thrones, Chapter 40, Cateyln