You have to slightly amend the macro \acroauthor
to get that right:
\begin{filecontents*}{\jobname.bib}
@article{oasis,
author={{\acroauthor{Organization for the Advancement of Structured Information Standards}{OASIS}}},
title={Some title},
journal={J. Something},
year={2012},
}
\end{filecontents*}
\documentclass{article}
\usepackage[authoryear]{natbib}
\usepackage{etoolbox}
\newif\ifabbreviation
\pretocmd{\thebibliography}{\global\abbreviationfalse}{}{}
\AtBeginDocument{\global\abbreviationfalse}
\DeclareRobustCommand\acroauthor[2]{%
\ifabbreviation #2\else #1\global\abbreviationtrue\fi}
\begin{document}
Here it is \citep{oasis}
A second time \citep{oasis}
\bibliographystyle{plainnat}
\bibliography{\jobname}
\end{document}
This will get the desired output.