Change font inside document to specific point size

Just tell LaTeX to use nine point type.

\documentclass[12pt]{article}
\usepackage{setspace,lipsum}
    \doublespacing
\begin{document}
\lipsum[2]

\begin{center}\setstretch{1}\fontsize{9}{11}\selectfont
\begin{tabular}{*3{l}} % <- this table should be set in 9pt
    Word & Word & Word\\
    Word & Word & Word\\
    Word & Word & Word\\
\end{tabular}
\end{center}

\lipsum[2]
\end{document}

For a really awful result, here's what you want:

\documentclass[12pt]{article}
\usepackage{setspace,lipsum}
    \doublespacing
\begin{document}
\lipsum[2]

{\strut\fontsize{9}{11}\selectfont
\begin{tabular}[t]{*3{l}} % <- this table should be set in 9pt
    Word & Word & Word\\
    Word & Word & Word\\
    Word & Word & Word\\
\end{tabular}}

\strut\lipsum[2]
\end{document}

The fact that in a table environment the interline spacing is reset to single spacing is a precise choice made by the setspace package. You can revert it by removing the setting the package does.

\documentclass[12pt]{article}
\usepackage{setspace,lipsum}
\usepackage[justification=centering, font=normalsize, labelfont=bf]{caption}

\makeatletter
\let\@xfloat\latex@xfloat % remove the redefinition made by setspace
\makeatother

\doublespacing

\begin{document}
\lipsum[2]

\begin{table}
\centering\fontsize{9}{11}\selectfont
\begin{tabular}{*3{l}}
    Word & Word & Word\\
    Word & Word & Word\\
    Word & Word & Word\\
\end{tabular}
\caption{My table}
\end{table}

\lipsum[2]
\end{document}