This is a pretty controversial topic. When the Wilderness Act was passed in 1964, there were not really any mountain bikes. The Act clearly bans motorized recreation, but it is somewhatunclear about muscle powered mechanical. I think it is clear enough that the judgment is against bikes.
The regulations written to flesh out the Act prohibited mountain bikes. The government’s interpretation has been very purist. Some really like that; others don’t; but it surely makes it harder to pass a Wilderness Bill.
Some mountain bike organizations have supported Wilderness anyway, but others have teamed up politically with the real target of the Wilderness Act, motorcycles, ATVs (although they didn’t exist in 1964), 4 x 4s, aircraft (except for some grandfathered backcountry airstrips), motorboats.
Allowing mountain bikes will probably require an explicit statement that they are allowed in future legislation.
Wild Bill: Are We Ready for Wilderness Lite? By Bill Schneider. New West.
Here is the text of the Act regarding “prohibited uses,”
(c) Except as specifically provided for in this chapter, and subject to existing private rights, there shall be no commercial enterprise and no permanent road within any wilderness area designated by this Act and, except as necessary to meet minimum requirements for the administration of the area for the purpose of this Act (including measures required in emergencies involving the health and safety of persons within the area), there shall be no temporary road, no use of motor vehicles, motorized equipment or motorboats, no landing of aircraft, no other form of mechanical transport, and no structure or installation within any such area.