Catching and spaying a stray cat?

I have a stray female cat living on my property. She is starting to warm up a little to me she will let me pet her while eating but not pick her up and she extremely shy and will run away sometimes. She was a kitten when we first discovered her on our property (really big property). She had one litter of kittens herself that’s how we found out she is female we found homes for the kittens and kept one who is very friendly with us. I want to get her fixed so she doesn’t keep having kittens but how will the vet handle fixing a stray cat? Will he put her to sleep so he can fix her since she will be freakin out and maybe clawing from the second we catch her and bring her in.