You can use fileext=lof
as an option for \DeclareFloatingEnvironment
so the supplementary figures will use the same auxiliary file than the regular figures; this will produce the inclusion in the LoF, but this might be confusing since both counters are independent:
\documentclass[a4paper,10pt]{article}
\usepackage{newfloat}
\usepackage{newfloat}
\DeclareFloatingEnvironment[name={Supplementary Figure},fileext=lof]{suppfigure}
\begin{document}
\listoffigures
\clearpage
\begin{figure}
\caption{Test figure one}
\end{figure}
\begin{suppfigure}
\caption{Test supplementary figure one}
\end{suppfigure}
\begin{figure}
\caption{Test figure two}
\end{figure}
\begin{suppfigure}
\caption{Test supplementary figure two}
\end{suppfigure}
\end{document}

A better option would be to use \listofsuppfigures
to produce a separate name; the name for this list can be changed using listname=
:
\documentclass[a4paper,10pt]{article}
\usepackage{newfloat}
\usepackage{newfloat}
\DeclareFloatingEnvironment[name={Supplementary Figure},fileext=lsf,listname={List of Supplementary Figures}]{suppfigure}
\begin{document}
\listoffigures
\listofsuppfigures
\clearpage
\begin{figure}
\caption{Test figure one}
\end{figure}
\begin{suppfigure}
\caption{Test supplementary figure one}
\end{suppfigure}
\begin{figure}
\caption{Test figure two}
\end{figure}
\begin{suppfigure}
\caption{Test supplementary figure two}
\end{suppfigure}
\end{document}
