Swansea have already knocked Premier League champions Manchester United out of this season's FA Cup and can reach the quarter-finals with a win at Goodison Park on Sunday.
Having also won the Capital One Cup last term and beaten Valencia away from home en route to the last 32 of the Europa League, Vorm admits there is a feeling at Swansea that they can beat anyone.
"We've already shown in the cups that everything is possible. We've had some of our best results in the cups and we'll go to Everton with a lot of confidence," the Dutch goalkeeper said in the Daily Mirror.
"Everton play some amazing football under Roberto Martinez but I think that suits us better and the way we play than, for example, going to Stoke.
"We seem to do well in the knock-out competitions so we won't have any fears. I think of all the wins we've had, Valencia away is the best. A top team, great trip for the fans and to win there for our fans was amazing."
Swansea have struggled since that victory at Valencia in September, with Michael Laudrup sacked as manager and Garry Monk picking up the reins.
Vorm is convinced that Swansea can get back on track and enjoy a successful finish to the season, although he still has sympathy for Laudrup.
He explained: "It was hard to say goodbye with Michael because we were all together for 18 months, we had some great times, won a trophy and got into Europe.
"The results weren't good enough this season. I've been a player long enough to know that it is always the manager that pays the price. I had great times with Michael, so it was a hard time.
"But I know Garry well. When I was coming back from my knee injury, we were in the gym together a lot. He was always going to be a manager, he's such a natural.
"The players know him, they like him, they respect him and they listen to him. He's no longer one of the boys, he's the gaffer. I told him straight away, 'Don't worry, we've got your back.' We are all behind him."
Vorm, who has been impressed by the performances of his fellow keepers in the Premier League, believes he can go to the World Cup this summer as Holland's No.1.
He said: "If you look at the table, Cardiff are second from bottom and yet their keeper, David Marshall, has been their best player. Fulham are bottom but I know just how good [fellow Dutchman] Maarten Stekelenburg is. Those are the keepers at the bottom, which shows you just how good the standard is.
"I think it pushes you as a keeper. It's competition to be the best.
"I went to the World Cup in 2010 in a different situation. But now I've played a lot of matches for the national team and I feel if I'm playing well for Swansea then I stand a good chance of being in the starting XI."