WASHINGTON—Legislation to remove regulatory obstacles to the development of self-driving vehicles is running into problems in the Senate, dimming prospects for the quick passage that many had expected.

The bill, known as AV Start, would set out a light-touch system of federal oversight of autonomous vehicles. Developers would be able to test and even market the vehicles long before detailed federal safety regulations are adopted—a process expected to take many years. The bill would also block states from adopting their own...