According to the Christian Post Breanna Koski and Joanna Duka, who run Brush and Nib Studio in Phoenix, Arizona, have been accused of violating the city's anti-discrimination laws by refusing to make the invites for a client.
The charity has said Ms Koski and Ms Duka have now been threatened with a £1,965 fine or six months in prison - the maximum sentences for breaking Phoenix non-discrimination rules.
Breanna Koski and Joanna Duka's request for a preliminary injunction - a legal attempt to temporarily stop a civil case from starting - was rejected by Superior Court judge Karen Mullins, who called the attempt "absurd" according to Christian Post. The pair are considering an appeal and are being supported by Christian advocacy charity ADF.
Jonathan Scruggs, from ADF, said: "No American, including artists, should have the government force them to create art against their artistic and religious beliefs."
In September a person named Zack Z left a comment about Brush and Nib Studio on the Yelp customer review website, saying: "I wanted Brush & Nib Studio to make wedding invitations for me but was told by the owner Breanna that they don't cater to gay weddings.
"This is a horribly bigoted position to hold in this modern day, and shocking beyond belief.
"I would advise others to avoid this business and the ugly attitude of the owners. Go elsewhere."
In response, Breanna Broski disagreed and said on the website: "Hi Zack! Unfortunately, we have never had the pleasure of speaking with you.
"Also, according to our records, we have never created or been asked to create any custom artwork for you.
"We regret that you have this opinion of our business, but respect your right to express your feelings. We wish you the best."