These types of vehicles are addressed in state law as "wheeled all-terrain vehicles" (WATV). RCW 46.09.310(19) defines "Wheeled All-Terrain Vehicle" as:
(a) any motorized nonhighway vehicle with handlebars that is fifty inches or less in width, has a seat height of at least twenty inches, weighs less than one thousand five hundred pounds, and has four tires having a diameter of thirty inches or less, or (b) a utility-type vehicle designed for and capable of travel over designated roads that travels on four or more low-pressure tires of twenty psi or less, has a maximum width less than seventy-four inches, has a maximum weight less than two thousand pounds, has a wheelbase of one hundred ten inches or less, and satisfies at least one of the following: (i) Has a minimum width of fifty inches; (ii) has a minimum weight of at least nine hundred pounds; or (iii) has a wheelbase of over sixty-one inches.
With respect to their operation on city streets, RCW 46.09.455(1) provides, in relevant part:
A person may operate a wheeled all-terrain vehicle upon any public roadway of this state, not including nonhighway roads and trails, having a speed limit of thirty-five miles per hour or less subject to the following restrictions and requirements:
(d)(i) A person may not operate a wheeled all-terrain vehicle on a public roadway within the boundaries of a city or town, not including nonhighway roads and trails, unless the city or town by ordinance has approved the operation of wheeled all-terrain vehicles on city or town roadways, not including nonhighway roads and trails.
So, WATVs may not be operated on city streets unless the city has adopted an ordinance to specifically authorize their use and the streets on which they are allowed have a speed limit of 35 mph or less. RCW 46.09.455(1). If a city does authorize their use, it must on the main page of its website identify the public roadways on which WATVs are allowed. RCW 46.09.455(1)(d)(ii).
In addition, RCW 46.09.360, relating to regulation of WATVs by local governments, provides in part:
Notwithstanding any of the provisions of this chapter, any city, town, county, or other political subdivision of this state, or any state agency, may regulate the operation of nonhighway vehicles on public lands, waters, and other properties under its jurisdiction, and on streets, roads, or highways within its boundaries by adopting regulations or ordinances of its governing body, provided such regulations are not less stringent than the provisions of this chapter.
This means the state has not entirely preempted the field and cities can enact additional regulations governing WATVs as long as the regulations are not less stringent than the state regulations.
For more information, see our July 23, 2013 MRSC Insight blog post, All-Terrain Vehicles Renamed and Rolling, and the Final Bill Report for ESHB 1632.
Although we have not surveyed cities on this topic, we are aware of a number that have adopted ordinances to allow and regulate wheeled all-terrain vehicles within their city limits. See, for example:
(Link to this question)