I figured out a solution, but please, feel free to pose a more elegant solution.
I needed to 1) reset the section counter at the appendix, 2) change the numbering from \Roman to \Alph , and 3) add a counter to the appendix. Here is a minimal example:
\documentclass{ieeeconf}
\usepackage{amsthm}
\newtheorem{theorem}{Theorem}
\begin{document}
\section{First Section}
\begin{theorem}
My theorem.
\end{theorem}
\section{Last Section}
\begin{theorem}
My other theorem.
\end{theorem}
\setcounter{section}{1}
\renewcommand{\thesection}{\Alph{section}}
\counterwithin{theorem}{section}
\section*{APPENDIX}
\begin{theorem}
Theorem in appendix.
\end{theorem}
\begin{theorem}
Another theorem in the appendix.
\end{theorem}
\end{document}
Hopefully, this is helpful to someone.