Can new customers get a discount automatically when they sign up?

Yes. Go to Marketing → Welcome Coupon:

1. Tick Send a welcome coupon when someone signs up.
2. Choose the Discount type (Percentage (%) or Fixed amount) and the value.
3. Optionally set a Minimum cart value, and set Valid for (days) (30 by default).
4. Leave Only usable on their first order ticked if it is meant for the first purchase.
5. Click Save.

When a shopper creates an account, WCart issues them their own single-use code (it starts with WELCOME), locked to the email they signed up with, and emails it to them. Nothing is sent until someone actually signs up, and a problem with the coupon or email never stops their account being created.