Autogyro and helicopter blades are more or less flexible, and when not stiffened by centrifugal force they droop noticeably. A extreme case would be that of fabric blades. They have been used at least once, as propeller blades in the airship ‘Parseval’. Could fabric blades work as rotor blades of an autogyro?
For thrust to pull on the hub, a propeller blade must be able to transfer shear loads from the circumference to the center. Add a bending moment when the distance between the circumference and the hub becomes nonzero. This requires a stiff propeller.
With flexible blades thrust can be achieved when the blades form a cone from the lift force pulling them forward. This will be added to the tension created by the centrifugal force on the blades. From the right picture it seems that no such lift was achieved when the picture was taken.
However, the Parseval dirigibles could achieve speeds above 40 km/h, so the caption "… UND IN FAHRT" (… and in motion) is not quite correct. The complex hub arrangement actually allowed to change blade pitch and even to reverse pitch in flight.
Austrian military dirigible M1 (picture source), a Parseval-type dirigible built in 1909.
Now to your question:
Could fabric blades work as rotor blades of an autogyro?
Maybe, when spun up before take-off. A typical autogiro does not need to power its rotor directly because it is set in motion by the airflow. Later Cierva autogiros did power the rotor in order to shorten their takeoff distance. Once in flight, all power is used to spin the propeller because driving the rotor would create undesirable torque which needs to be counteracted either by a second rotor spinning in opposite direction or by a tail rotor like in Sikorsky-type helicopters.
It is important to keep the blades at a positive angle of attack so they will constantly produce enough lift to keep the autogiro afloat. While it seems that changing the attachment points of the two ropes which formed the leading and trailing edges of the propeller blades was enough for the Parseval propeller, the much higher aspect ratio of a typical autogiro blade will most likely restrict lift production to the blade section close to the root. Since a flat plate is indifferent to pitch changes, the pitch angle of such a blade would be only controlled by the tension in the ropes from the centrifugal force. This could work but I would rather use a stiff blade to make sure that their angle of attack is well controlled.
Also, a good airfoil will increase the range of possible angles of attack with good efficiency which is especially helpful for helicopters. Using a flat plate airfoil will definitely cost performance.