(I cannot for the life of me draw backgrounds, they're the bane of my existence fhjkdchkvndk)
Emily and Peter are not, in fact, Rabbit's biological children.
As I kept watching the show I noticed some details couldn't simply add up if the twins were Rabbit's bio children. In "Lucy Blows her Top", Lucy muzes that, as one year and a half babies, they shouldn't already be as agitated as they are. In the same episode, Wolf confirms to us, the audience, that its has been canonically around a year since he and his sons moved to England. Which means when the first episode aired, the twins were only at most six months olds.
All of this doesn't add up with what we learn about Rabbit's late wife in "Barry Loves Lucy", in which we discover he's been lying to Lucy about what happened to her mother. She would not have buy that her mother stayed in Canada to live in a moose farm unless she was very young when Rabbit told her, and thus more naive (probably around three, four years old). Plus, in "Action Rabbit", he indicates to Terrapin that Lucy "grew up without a strong female figure". Once again, it points out that it has been years since her mother passed away. It would not make sense otherwise to use those terms if only six months had passed until the moment the first episode takes place.
But the one little bit that actually fed that headcanon the most is how Lucy asks about her mother:
"Dad, where is my mother?"
Not mum. Not our mother. My mother.
So, combine all of this aaaaand the new question is why in the world would Rabbit, a widower and single father, adopt two baby rabbits (especially when he already struggles to connect with his daughter) ?
My headcanon is that this is a kinship adoption, one that Rabbit reluctantly agreed to because the only other option would be to send the twins to foster care (which, in his opinion, was even worse). I'm working on how exactly Rabbit and Lucy are biologically related to them, but I like to think Rabbit is a cousin to one of their bio parents.
(There may be a comic coming to expend a bit on all of this...)