It is difficult to pick out who would be the antagonist, as it isn't a typical good guy vs. bad guy story. But one can certainly point to the two protagonists, Woodrow and Gypsy.
They both act as protagonists since they are the main focus of the good and powerful emotional aspects of the story. They serve to highlight the depth of love and caring for someone without some of the usual surface attractions that many relationships are based on.
If you had to find an antagonist, you could perhaps point to Belle as she has a relatively superficial and harmful competition with Gypsy, basically because she isn't as beautiful as Gypsy is so she feels she has to in some way bring her down.