Sheep vs. Furry animals
Fur, wool, and hair are all made of keratins. 

To the best of my knowledge wool and fur are separated arbitrarily, based on the properties of the fibres. This arbitrary division allows rabbits to have fur but selective breeding has produced angora rabbits, which have wool. The opposite should be possible, with time you could breed a sheep that has fur. 

Length; Wool grows continuously and animals are clipped when the fibres are long. Fur often has a maximum length.

Structure: The structure of the keratin in the wool makes the fibres kinked (crimped). These kinks aid the production of fabrics. I believe the cuticular scales also aid in the production of fabrics.



Fur doesn't have kinks, so even if it's long enough to weave the fabric is likely to lack durability. 

Some furs are used in textiles rather than pelts. Possum fur is 'plucked' and woven into fabrics with wool. The resulting fabric is slightly less durable. However, possum fur is hollow and smooth so possum fur fabrics are very soft and warm. The clothing is very good to to wear. 

However, fur is generally inferior to wool for textiles and it took a couple of decades to develop techniques to produce textiles from possum fur. This happened because of odd ecological and economic drivers. Possums are are a major pest species in New Zealand where they contribute to the decline of native birds and insects. Millions of possums are killed every year in New Zealand. The market for possum skins is very small. This is why so much effort was put into producing a fabric from possum fur. 

So you could shear a fox and produce a fabric but the fabric would likely be poor and there's no reason to do so.