Olivia here from the Shopify Community team.
I appreciate you reaching out with your question. I assume you might be referring to this subscribe section on the Coming Soon page:
The reason I ask is this is sometimes confused with the log-in section from that same landing page. If not, then we would expect all subscriber email addresses to show up in your Shopify Admin > Customers section. Typically, the only reason they may not could be narrowed down to two issues:
A. It is not a Shopify-developed theme
It's totally okay to use themes that are not listed on our Theme Store. However, as they may not have been built specifically for the Shopify platform, this could cause some hiccups. In these cases, we aren't able to troubleshoot third-party code, however, your original Theme Developer should be able to help out. Make sure to contact the support team where you purchased your theme for further support.
B. Custom scripts have been injected into the theme
This is unlikely unless functionality was edited through an injected script, in which case you will want to revert back to an original version of your theme. You can download up to 20 back-ups in your Shopify Admin > Online Store > Themes section.
This isn't to say that you can't use other themes or add custom scripts - just that there may be a command that isn't specific to one Shopify might recognize. If your issue doesn't fall into one of these two options, then I will want to take a deeper look at your store with you. Please let me know and I can send an email to you with next steps.
Welcome to Shopify Community,