\documentclass{exam}
\usepackage{graphbox,wrapfig,duckuments,blindtext}

\usepackage{xparse}

\NewDocumentEnvironment{mywrapfigure}{O{}mO{\wrapoverhang}mO{1pt}}{%
    \setlength{\intextsep}{#5}%  <<< has no effect
    \wrapfigure[#1]{#2}[#3]{#4}%
}{%
    \endwrapfigure%
}

\begin{document}
    \begin{mywrapfigure}{r}{0.3\linewidth}
        \includegraphics[width=\linewidth]{example-image-a}
    \end{mywrapfigure}
    \blindduck[1]
    \bigskip
    \begin{questions}
        \begin{minipage}[t]{\dimexpr \linewidth}
            \question%\the\everypar
            \begin{mywrapfigure}{r}{0.3\linewidth}
                \vskip-\baselineskip
                \includegraphics[width=\linewidth]{example-image-b}
            \end{mywrapfigure}
            \blindtext[1]
        \end{minipage}
        \bigskip%\hrule
        \begin{parts}
            \begin{minipage}[t]{\linewidth}
                \part[4]%\the\everypar
                \begin{mywrapfigure}{r}{0.3\linewidth}
                    \vskip-\baselineskip
                    \includegraphics[width=\linewidth]{example-image-c}
                \end{mywrapfigure}
                \blindduck[1]
            \end{minipage}
        \end{parts}
    \end{questions}
\end{document}