Samantha Henderson, 25, from Corfe Castle, Dorset was reported missing just before 08:45 GMT on Thursday.
A 27-year-old man from the village, who is known to Ms Henderson, has been arrested on suspicion of her murder.
A statement released by her family said her disappearance was "completely out of character" and appealed for information.
"She is hard-working, always presentable. She would never walk out and leave her children," it said.
"If anyone knows where she is or what has happened, please contact the police."
They also made a direct appeal to Ms Henderson, saying: "No one is upset or thinks anything badly of you Sam.
"We just want you to come home as everyone wants to help and support you."
Ms Henderson had last been seen leaving her home at 16:30 on Wednesday.
Extensive police investigations are continuing, including a forensic examination of her house and the surrounding area.
The police helicopter has also been used in the search.
Det Insp Neil Devoto said police had "grave concerns" for her and appealed for information from anyone who may have seen her or had contact with her on social media.
"While we are hoping for the best, sadly we believe she may have come to some harm," he said.
"She has never gone missing before and, from what we understand, it is almost unfathomable that she would have left her four young children."
Friends of Ms Henderson have been distributing posters appealing for information about the missing woman.
She is described as white, 1.60 metres (5ft 3 inches), of a very slim build and with blonde hair.
She was seen wearing a grey jumper, blue jeans and slippers.
The family of a mother-of-four whose disappearance sparked a murder inquiry have made a plea for her to come home.