Question

Which group of people disputes both the land-bridge and coastal-route theories?

Answer 1

Not everyone agrees with both the land-bridge and coastal-route theories. But in all the group of people, many Native Americans dispute the land-bridge and coastal-route theories. Everyone has their own tradition of explaining how they settled in their lands. 

Answer 2

These theories try to explain how and when the human race first arrived to America.

The more accepted theory is the Land-bridge one, this theory is based on the idea that a "brige" formed  between Asia and North America thanks to the last ice age, this allowed humans to travel and settle there between 40,000 and 15,000 years ago.

Another theory is the Coastal-route one, it states that the first americans arrived by boat.  They traveled down the Pacific coast, living in areas of ice-free land, called refugia.

Therefore the people disputing these theories are scientists, most of them geologists and archaeologists. To support the theories they need to analize the ground and the artifacts used by the first settlers.