indeed gut of the sheep used for making violin and all stringed instrument strings,but I think we do have to look at this quote in context :-)

The Vicar is talking of clothing:

When Sunday came, it was indeed a day of finery, which all my
sumptuary edicts could not restrain. How well so ever I fancied
my lectures against pride had conquered the vanity of my
daughters; yet I still found them secretly attached to all their
former finery: they still loved laces, ribbands, bugles and
catgut; my wife herself retained a passion for her crimson
paduasoy, because I formerly happened to say it became her.

I must confess I can't see these ladies draping themselves with bits of violin strings qualifiying as "finery" !LOL

Catgut was, in the mid 18th century , a term used for plain cotton canvas that was used as a base for detailed emboridery.

I think here the Vicar's wife and daughters had proabably a pocket or some part of their clothing,a stomacher perhaps, that was made of cotton canvas richly embroidered.

The stomacher was a stiff piece of fabric used to cover the opening of the bodice of type of dresses worn at this time.I'll put a photo up later ( with Myretta's kind assistance!) so you can all see what I mean.

If anyone is interested, paduasoy ,as mentioned in the above passage,is a corded silk fabric, that originalted from Padua in Italy,hence its name.(In France it was called soie silk or pou-de-soie.