There a wide-variety of email marketing apps available in our Shopify app store that would do the trick. Depending on how big your customer size is, there may be certain apps that are better than others in terms of cost.
Seguno is free for up to 250 subscribers
Jilt is free for up to 600 sends per month
If there is anything else I can help you with, please let me know.
If you have thousands of customers sending an email all at once will not be possible. We could build you a custom app and have you test something out in a day or two. Do you already have a mailserver? Is it sending emails using your custom domain?