Villa won Tuesday night's semi-final second leg 2-1 but crashed out 4-3 on aggregate as the League Two Bantams became the first first team from the fourth tier to reach a major Wembley final.
With Villa just a point above the Premier League drop zone, having failed to win any of their last six games, the pressure is now mounting on Lambert.
But the Scot, who quit Norwich to take over from Alex McLeish in the summer, insists he will not walk away from Villa Park.
When asked if he was still the man to take Villa forward, Lambert said: "Yes, absolutely. When you ask managers that, they will tell you they just get on with it until they hear anything different.
"You have to run with it. There are two ways. You either lie down and take it or you come out fighting.
"I am certainly not going to lie down. I have never done it. I will come out fighting."