Sarah West, 46, was last seen walking towards Filey on the A165 near Cayton Bay at around 8am on Sunday, April 25.
In a joint statement, Ms West's mother Valerie and sister Rebecca said: "Sarah, please come home. We love you and we are here for you no matter what.
"We know you are worried about your health and our health and we just want you to come home so we can support you. Together we’ll find the right treatment and the right people to help you and we will be by your side throughout it all.
"There are so many people who love you and care about you.
"We are all praying for you to come home, that’s all we want and we will be here waiting."
Police have appealed for motorists travelling along the A165 between Scarborough and Reighton to get in touch if they have dashcam footage from the road anytime after 7.30am on Sunday which may show Ms West.
Her family have thanked everything involved in the searches to find her.
They said: "Please keep vigilant for any sightings of Sarah and help us to bring her home safe. We are so grateful for the support of everyone who is helping to find her."
Ms West is 5ft 5ins tall and of a very slim build with dark blonde/brown medium length hair. She is believed to be wearing jeans and a blouse top.
Anyone with information should call 101 or 999 for an immediate sighting.