Plugin shortcodes not working in custom theme

As was pointed out by Michael, I had been missing the wp_head() and wp_footer() functions.

To anyone who might be in the same boat as I, you can have a look at the function references here (head) and here (footer).

Thank you, to anyone who took their time to help me out, even if all they did was read the question. I’m glad we reached a conclusion so quickly!