As far as I am aware, there is not an ASTM standard for fiber contact angle measurements. But you have several approaches for measuring this.
1. Sessile drop. Depending on the diameter of your fiber, you may need advanced equipment for this. Typical contact angle apparatuses dispense droplets in the few-uL volume range, and the droplets will have a diameter of some mm depending on how much they spread. For fibers, these dimensions are usually too large. Instead, a system with better optics and a pL droplet dispenser is used. These droplets have diameters on the order of microns. To get a good image of the droplet, cut the fiber to a very small piece (a few mm maybe?), and secure it onto a glass slide, maybe with glue; make sure the fiber is not tilted. Orient the fiber so its long axis is along the camera-light source axis. Modern contact angle software packages allow for circular baseline (i.e., your fiber) corrections to the droplet shape fit and contact angle determination. If your fibers are very hydrophobic, you may have some trouble with the droplets rolling off.
2. Single fiber dipping. Using a force tensiometer with a sensitive mass balance, "dip and retract" the fiber from your liquid by having the instrument move the liquid container up and down while your fiber is hooked, stationary, to the mass balance. This gives you the advancing and receding contact angles.
3. Washburn method. The same principle applies for powder wettability measurements described here: https://www.biolinscientific.com/measurements/powder-wettability. The difference is that instead of powders, you're packing fibers. Typically the control solvent for these measurements is hexane, so you need to make sure that hexane won't dissolve your fibers if you go this route. This method can be dependent on operator handling to some extent, particularly with how you pack the material inside the cylinder, so be very careful and consistent if you use this method. Contact angle error on the order of a few percent is considered good when using the Washburn method.
I would probably recommend you try the second method if it's available to you.