Can the government acquire or “take” private land within the river corridor?

Yes, but with many restrictions. It is important to note that condemnation is a tool that has been used only rarely on wild and scenic rivers. The objective of wild and scenic river designation is to protect and, as possible, enhance the values which caused the river to be designated. Should some proposed or actual use clearly threaten the values the river was designated to protect, the river managing agency would work with a landowner to explore ways to avert the threat through local zoning, state provisions, land exchanges, or purchases on a willing-seller/willing-buyer basis. Condemnation would be a last resort, would only be feasible if funding were available, and is prohibited on some wild and scenic rivers by their enabling legislation.

Source
Interagency Wild & Scenic Rivers Council