The McArthur family, who run Ashers Baking Company, are seeking to overturn a judgment which found they acted unlawfully by declining the order in May 2014.
Daniel McArthur, 26, Ashers' general manager said they hoped it would be their last court appearance.
"The County Court judgment was disappointing but we're hoping and praying that these judges will recognise what's at stake and decide to protect our freedom to express our Christian beliefs."
The Northern Ireland Equality Commission, which monitors compliance with the region's anti-discrimination laws, brought the landmark civil action against Ashers on behalf of a gay rights activist Gareth Lee.
The high profile case was heard at Belfast County Court over three days last March.
Mr Lee, a member of the LGBT advocacy group Queer Space, had wanted a cake featuring Sesame Street puppets Bert and Ernie with the phrase Support Gay Marriage for a private function marking International Day Against Homophobia.
He paid the £36.50 in full at Ashers' Belfast city centre branch but was telephoned two days later and told the company could not fulfil his order.
In evidence Ashers' owner Karen McArthur said as a born again Christian, she knew in her heart she could not make the cake (below) but had taken the order to avoid a confrontation in the shop.
Daniel McArthur also told the court his family could not compromise their religious beliefs, despite the legal ramifications.
Mr Lee claimed he was left feeling like a lesser person.
Delivering her findings District Judge Isobel Brownlie said the bakers had breached equality legislation and directly discriminated against Mr Lee, contrary to the law.
Ordering Ashers to pay agreed damages of £500, the judge said religious beliefs could not dictate the law.
Ashers Baking Company has six branches, employs over 80 people and delivers across the UK and Ireland.
Throughout the legal battle they have been supported by The Christian Institute which has organised public rallies and garnered financial backing for the case.
The Equality Commission has declined to comment ahead of the appeal.
David Smyth, from the Evangelical Alliance Northern Ireland, told Premier's News Hour Asher's argument: "There's no discrimination in this case because they wouldn't have provided this cake with this message to anyone... so if the Equality Commissioner himself would have walked in and ordered this cake he would not have been been provided with it, because of the message on the cake and not because of the sexual orientation of the customer."
He said: "It's being refused a specific message with which the owner profoundly disagrees."
Mr Smyth said that many traditional views can face opposition in wider society: "We are seeing some level of opposition to that creeping in, and I suppose that's our fear - it's not about Christians trying to seek special protection in any way, it's trying to maintain a free and fair public square with different views to be held."
Listen to Premier's Aaron James speak to David Smyth here:
Listen to Premier's Alex Williams speaking to Simon Calvert from the Christian Institute:
