For anyone else coming this way who doesn’t want to re-install their DSM, I found another solution that works (or did for me).
If you follow the steps in the previously linked guide (https://www.reddit.com/r/synology/comments/6a27sd/step_by_step_guide_to_installing_tiny_tiny_rss/) you’ll be using php56 for everything. What I found was that despite the php settings panel saying it was installing the openssl, curl, and sockets modules for php56, they were going in the base php modules directory and configuration. What I did to fix this:
sudo cp /usr/lib/php/modules/openssl.so /usr/local/lib/php56/modules/
sudo cp /usr/lib/php/modules/sockets.so /usr/local/lib/php56/modules/
sudo cp /usr/lib/php/modules/curl.so /usr/local/lib/php56/modules/
Then open up /usr/local/etc/php56/php.ini and add:
extension = openssl.so
extension = curl.so
extension = sockets.so
And that should hopefully fix it. Don’t know for sure that curl and sockets are needed, but seemed relevant so I added them.