That's weird that your Apache SSL settings would have any bearing on PHP and egress connections. Glad it works though, thanks for the update! Best guess is that maybe you were missing OpenSSL, and setting up Apache correctly got you the necessary libraries.
It appears adding TLS support properly in my Apache config worked. To those who are using a setup like mine:
LAMP stack on CentOS 7
Used @Blesta.Store's tutorial (https://licensecart.com/brain/knowledgebase/398/How-to-install-Blesta-on-CentOS-without-a-panel.html)
Let's Encrypt SSL
Apache vHost (which you need anyway to use Let's Encrypt)
Your Apache config file needs to look like so:
<VirtualHost *:80>
ServerAdmin email@yourdomain.com
DocumentRoot "/var/www/html/yourdomain.com/"
ServerName yourdomain.com
ServerAlias www.yourdomain.com
ErrorLog "/var/log/httpd/yourdomain.com-error_log"
CustomLog "/var/log/httpd/yourdomain.com-access_log" combined
<Directory "/var/www/html/yourdomain.com/">
DirectoryIndex index.html index.php
Options Indexes MultiViews FollowSymLinks
AllowOverride All
Require all granted
</Directory>
Redirect permanent / httpd://yourdomain.com/
</VirtualHost>
<VirtualHost *:443>
ServerAdmin email@yourdomain.com
DocumentRoot "/var/www/html/yourdomain.com/"
ServerName yourdomain.com
ServerAlias www.yourdomain.com
ErrorLog "/var/log/httpd/yourdomain.com-error_log"
CustomLog "/var/log/httpd/yourdomain.com-access_log" combined
<If "%{HTTP_HOST} == 'www.yourdomain.com'">
Redirect permanent / https://yourdomain.com/
</If>
<Directory "/var/www/html/yourdomain.com/">
DirectoryIndex index.html index.php
Options Indexes MultiViews FollowSymLinks
AllowOverride All
Require all granted
</Directory>
SSLEngine On
SSLProtocol -ALL +TLSv1.1 +TLSv1.2
SSLCertificateFile /etc/letsencrypt/live/yourdomain.com/cert.dh.pem
SSLCertificateKeyFile /etc/letsencrypt/live/yourdomain.com/privkey.pem
SSLCertificateChainFile /etc/letsencrypt/live/yourdomain.com/chain.pem
</VirtualHost>
SSLProtocol -ALL +TLSv1.1 +TLSv1.2 is what's vital in this situation I was having.