Can chiggers spread by scratching?

No: It just makes the itching worse!

Not likely: Chiggers to not burrow into the skin. They get in a hair follicle put a proboscis in to the skin and emit a substance into the skin that makes a solution they eat. It is the substance that causes the irritation. Scratching does not make them spread. There are many home remedies to get rid of them. Rubbing alcohol, cover with nail polish, etc.