When you have a contractual relation with a customer, you have sufficient legal basis to send emails to your customer. Proof of contract can be given by any means including a written document or when the customer registers in a system. Even an oral contract can be enough.
Of course, the mails you send must contain information relating to the contract. Do not go off topic.
Feel free to check our GDPR Questions and Answers section for further GDPR related information.