Renata from Shopify here.
An email marketing app would be appropriate to send your customers newsletters or email the eligible ones with the discount code you have just created. Simply subscribing to email marketing via the email form from your store won't send out any emails automatically unless you have an app to do such thing.
We have some great options in our app store that's worth checking. I'll list some options below but you can also check the complete list of apps from our store here:
Once you're set up with your preferred app, you can then send the discount code created to a specific group of customers (in this case, the ones who subscribed to newsletter - emails subscribers).
More about this process can be seem in this topic from our Help Center.
Hope it helps!
I am quite new to shopify and find it amazing so many things like automated email has to be done through apps. I have spent days working out why I cannot get an email as a newsletter subscriber. You have them for new customers, abandoned checkout and many others but not for newsletter subscribers.... I am close to ditching shopify as a new business startup because to make it fully functional (which you say it is) you have to add quite a few apps which cost more and more again. So much for $29 a month. And yes so many apps offer "for free" service but with bugger all features enabled. Looks to me like shopify itself only half completed what could be an awesome site builder to then give up and throw it out to independent developers. Not something I will ever recommend to other site developers from what I have experienced so far.