If/Else Statement Not Working When Trying To Reformat Subsections

Your \ifnum is only evaluated once, precisely at a moment when subsection has value 0.

You need the conditional to be evaluated when \thesection is expanded:

\renewcommand{\thesection}{%
  \thesection.%
  \ifnum\value{subsection}<10
    0%
  \else
    \ifnum\value{subsection}<100
      0%
    \fi
  \fi
  \arabic{subsection}%
}

A different solution with xparse, which is easily generalizable to any (fixed) padding: we count the number of items in the expansion of \arabic{subsection} (that is, the number of digits) and pad accordingly with zeros, in this case to three digits.

\usepackage{xparse}

\ExplSyntaxOn
\RenewExpandableDocumentCommand{\thesubsection}{}
 {
  \thesection.
  \prg_replicate:nn { 3 - \tl_count:f { \arabic{subsection} } } { 0 }
  \arabic{subsection}
 }
\cs_generate_variant:Nn \tl_count:n { f }
\ExplSyntaxOff

Full example:

\documentclass[a4paper, 12pt]{article}
\usepackage{xparse}

\setcounter{section}{-1}
\renewcommand{\thesection}{Module \arabic{section}}

\ExplSyntaxOn
\RenewExpandableDocumentCommand{\thesubsection}{}
 {
  \thesection.
  \prg_replicate:nn { 3 - \tl_count:f { \arabic{subsection} } } { 0 }
  \arabic{subsection}
 }
\cs_generate_variant:Nn \tl_count:n { f }
\ExplSyntaxOff

\begin{document}

\title{Notes}
\author{}
\date{}
\maketitle

\section{Introduction}

\section{A}

\subsection{A1}

\subsection{A2}

\subsection{A3}

\subsection{A4}

\subsection{A5}

\subsection{A6}

\subsection{A7}

\subsection{A8}

\subsection{A9}

\subsection{A10}

\subsection{A11}

\setcounter{subsection}{99}

\subsection{A100}

\section{B}

\subsection{B1}

\subsection{B2}

\section{C}

\section{D}

\section{E}

\end{document}

The \if condition belongs into the macro definition of \thesubsection:

\renewcommand\thesubsection{\thesection.\ifnum\value{subsection}<10 0\fi\arabic{subsection}}