Often, yes. Camping outside a designated campground, which the U.S. Forest Service calls dispersed camping, is allowed on the vast majority of national forests and grasslands and is usually free, with no permit or reservation needed. The trade-off is that dispersed sites have no services such as water, toilets, or trash pickup.
There are exceptions. The Forest Service notes that heavily used areas, such as popular trailheads or some wilderness areas, may require a permit and fee, and developed campgrounds with facilities generally charge a recreation fee. Many forests also limit how long you can stay in one dispersed spot; on the Caribou-Targhee National Forest, for example, the limit is 14 days in any 30-day period. Check the local ranger district’s rules before you go.
Source: U.S. Forest Service, “No reservation? No problem!”; stay-limit example from the Caribou-Targhee National Forest dispersed camping page.
Image: Dispersed campsite in Manistee National Forest, Michigan, U.S. Forest Service photo by Savannah Halleaux (public domain, U.S. government work).