With biblatex-apa you only need
\renewbibmacro*{author}{%
\ifnameundef{author}
{\usebibmacro{labeltitle}}
{\printnames[apaauthor][-\value{listtotal}]{author}%
\setunit*{\addspace}%
\printfield{nameaddon}%
\ifnameundef{shortauthor}
{}
{\setunit*{\addspace}%
\printtext[brackets]{\printnames{shortauthor}}}%
\ifnameundef{with}
{}
{\setunit{}\addspace\mkbibparens{\printtext{\bibstring{with}\addspace}%
\printnames[apaauthor][-\value{listtotal}]{with}}
\setunit*{\addspace}}}%
\newunit\newblock%
\usebibmacro{labelyear+extrayear}}
Where we just added the part starting with \ifnameundef{shortauthor}.
MWE
\documentclass[british]{scrartcl}
\listfiles
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@book{iea,
author = {{International Energy Agency}},
title = {World Energy Outlook},
shortauthor = {IEA},
date = {2005}
}
\end{filecontents}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{babel,csquotes}
\usepackage[
style=apa,
backend=biber
]{biblatex}
\addbibresource{\jobname.bib}
\addbibresource{biblatex-examples.bib}
\DeclareLanguageMapping{british}{british-apa}
\renewbibmacro*{author}{%
\ifnameundef{author}
{\usebibmacro{labeltitle}}
{\printnames[apaauthor][-\value{listtotal}]{author}%
\setunit*{\addspace}%
\printfield{nameaddon}%
\ifnameundef{shortauthor}
{}
{\setunit*{\addspace}%
\printtext[brackets]{\printnames{shortauthor}}}%
\ifnameundef{with}
{}
{\setunit{}\addspace\mkbibparens{\printtext{\bibstring{with}\addspace}%
\printnames[apaauthor][-\value{listtotal}]{with}}
\setunit*{\addspace}}}%
\newunit\newblock%
\usebibmacro{labelyear+extrayear}}
\begin{document}
\cite{companion,iea}
\printbibliography
\end{document}
