Do any airports in the US provide a procedural (non-radar) service or has everywhere got radar?
In the US there are many non-towered airports that require no-radar procedures.
Lake Placid, NY (KLKP), offers two GPS approaches that must be flown as full procedure approaches and
Saranac Lake, NY (KSLK), an airport serviced by an airline, offers an ILS that must be flown as a full procedure ILS.
In terms of towered airports, I recall
Latrobe, Pennsylvania, not having radar services.
The
US Airport/Facilities Directory, which is now online (linked with the name there), has a visual indicator that will tell you whether an airport is within radar coverage or not. Locate an airport in the directory and then look to the left of the
APP/DEP CON heading. If there is an R with a circle around it that airport has radar coverage. No circled R, no coverage.