Villa crashed out at the semi-final stage over two legs, going down 4-3 on aggregate to League Two side Bradford City and continuing the poor run of form which has dropped them into a Premier League relegation battle.
Bradford fans chanted 'you're getting sacked in the morning' to Lambert during the second half but Vlaar says the squad have total faith in their boss and are backing him to turn things around.
"The players are still all behind the manager. That is not an issue. Why should it be?" Vlaar said.
"I didn't hear the Bradford fans sing that (about being sacked). I cannot believe that will happen. I believe he is the right man for the job.
"We have to pick ourselves up. We have no option. We have no choice. We are fighting really hard in the league.
"Everyone is very disappointed. We have to get ourselves together and start over again. Staying up is the most important thing for us. We have to look forward, keep our head, get together.
"We all know the situation we are in so we have to fight back. But we want to do well in the FA Cup as well if we can.
"The supporters were brilliant against Bradford. The atmosphere was really great. I can't remember this before in my time with Villa. It gave such a good feeling.
"I understand everyone is disappointed and upset. That's their right. We are also disappointed. I hope they stay behind us like they did every game this season. This is going to be hard for them and I understand that.
"They also know what situation we are in and we need their support and I really trust and believe that they will do that."