Why does a state need a waiver to set emission standards?
Mostly, to prevent costly, repetitive litigation. There are literally 100s of cases where state laws have been challenged in federal court for creating a burden on interstate commerce1; including, for example, mudguards on trucks (Bibb v. Navajo Freight Lines, 359 U.S. 520 (1959).) and the number of freight cars on trains (Southern Pacific Co. v. Arizona ex rel. Sullivan,325 U.S. 761 (1945).) These cases occurred in the absence of Federal regulation.
In the case of emissions, Congress passed the Clean Air Act of 1963 to establish uniform regulations for the United States.
It was first amended in 1965, by the Motor Vehicle Air Pollution Control Act, which authorized the federal government to set required standards for controlling the emission of pollutants from certain automobiles, beginning with the 1968 models.
In doing so Congress specifically preempted1 states from passing their own regulations in the absence of federal approval. The first waiver was granted in 1968 for model year 1969.
July 16, 1968, Volume 33, Number 137, pg. 10160 - Department of Health, Education, and Welfare; Office of the Secretary; Motor Vehicle Pollution Control; California State Standards; Waiver of Application of Section 208, Clean Air Act
The Clean Air Act allows California to seek a waiver of the preemption which prohibits states from enacting emission standards for new motor vehicles. EPA must grant a waiver, however, before California’s rules may be enforced. When California files a waiver request, EPA publishes a notice for public hearing and written comment in the Federal Register. The written comment period remains open for a period of time after the public hearing. Once the comment period expires, EPA reviews the comments and the Administrator determines whether the requirements for obtaining a waiver have been met.
States and counties have the power to enforce safety and emission inspections, vehicle noise ordinances, no-idling laws, etc.
Not entirely. Vehicle inspection in the United States:
In the United States, vehicle safety inspection and emissions inspection are governed by each state individually. ...
The Clean Air Act of 1990 required some states to enact vehicle emissions inspection programs.
The EPA monitors Anti-Idling Regulations and may choose to act if they find a reason.
On vehicle noise, see Noise Pollution and Abatement Act of 1972:
The Act established mechanisms of setting emission standards for virtually every source of noise, including motor vehicles, ...
Congress ended funding of the federal noise control program in 1981, which curtailed development of further national regulations. Since then, starting in 1982, the primary responsibility to addressing noise pollution shifted to state and local governments.
1 Mentioned in a comment.