<...And I don't know how to find out.

Well, of course I do ;). This site gives a lot of the history of tapestry in general, showing its use in many civilizations. Of interest is this paragraph:

However, the use of tapestries died out in the West and was not reintroduced until the Moors established a civilisation in Spain during the eighth century. The industry spread from Spain to France and by the eleventh century Poitiers, Arras and, later, Tournai were renowned in Europe for the tapestries produced there. Flanders had all the ingredients needed for textile production; a wealth of fuller's earth, plants suitable for dyestuffs, access to superior English wool and a plentiful water supply from the surrounding rivers. Gradually, the craft became established in Ghent, Bruges, Ypres and Valenciennes also. Flanders became regarded as the chief tapestry weaving area for three centuries with Antwerp serving as the European tapestry market.

Although the source does not seem to be the Crusades specifically, it does point to an initial Muslim influence.