On the Facebook post, Ms Gray said the twins had an extremely rare endocrine condition, which meant they had been "in and out" of hospital for nearly seven years.
She said her daughter, and grandsons had used the service in the past without any problems.
She thanked the good Samaritan who offered her family a ride and delivered them to Wellington Hospital, safe and sound.
"My appreciation and thanks is extended to the kind motorist who refused a contribution towards their petrol cost and went out of their way to deliver them safely."
Porirua mayor and Capital and Coast District Health Board (CCDHB) member Nick Leggett, said he had been concerned by the post.
"I was really concerned. The idea that kids are excluded from hospital treatment because their parents haven't got a child safety seat is a worry.
"We cater a lot of services around people with particular needs, children often have needs and for no fault of their own, they need either their parents or the system to be able to provide for them.
"In this case its seems like a pretty simple and sensible thing to do. The principle is we should be thinking about this for everything we do for kids."
Executive director, clinical and support services Gina Lomax said she had spoken with Tanyia's daughter, apologised and assured her that the CCDHB would ensure she and her children were able to travel on the shuttle in future.
"Our driver was following our shuttle conditions, which states parents must supply an appropriate child restraint for children aged seven or under."
She also confirmed the board would now be purchasing booster seats.
"...So if parents don't have suitable restraints for school aged children they will be able to safely travel in our shuttles.
"We will also clarify our shuttle conditions with transport staff to ensure a consistent service for all users," Ms Lomax said.
