It is not exactly clear what is desired, from the question, so I will make an assumption that it is the yellow text the OP wants centered in the TOC, but not in the document. Also, how to handle multi-line entries of the chapter name in such a fashion? For the general approach, I surrounded the heading TOC text with \hfills, as in
\addcontentsline{toc}{chapter}{%
\ifHeadingNumbered
\mw@seccntformat{\color{ForestGreen}\@chapapp\ \HeadingNumber}%
\fi
\color{yellow}\hfill\HeadingTOCText\hfill}%
However, that does not handle the multi-line chapter names. For those, I require manual breaks using \manbreak, as in \chapter{This is a really extra long \manbreak chapter name}. This macro is defined one way for the TOC construction, and \relaxed for the rest of the document by way of
\def\manbreak{\hfill\hfill\\\mbox{}\hfill\hfill}
\tableofcontents
\let\manbreak\relax
Here is the MWE.
\documentclass{mwbk}
\usepackage[utf8]{inputenc}
\usepackage{mathptmx}
\usepackage{graphicx}
\usepackage[dvipsnames]{xcolor}
\usepackage{blindtext}
\makeatletter
\renewcommand*\l@chapter[2]{%
\ifnum \c@tocdepth >\m@ne
\addpenalty{-\@highpenalty}%
\vskip 1.0em \@plus\p@
\setlength\@tempdima{1.5em}%
\begingroup
\parindent \z@ \rightskip \@pnumwidth
\parfillskip -\@pnumwidth
\leavevmode \bfseries \LARGE % added for large font for chapters
\advance\leftskip\@tempdima
\hskip -\leftskip %\@chapapp~ \par % added \@chapapp~ \par
#1\nobreak \raggedright % originally \hfil
%added for leaders (dots) in chapter in toc
%\xleaders\hbox{$\m@th
% \mkern \@dotsep mu\hbox{.}\mkern \@dotsep
% mu$}\hfill%
%%%
\nobreak\hb@xt@\@pnumwidth{\hss \itshape \fbox{#2}}\par\vspace{12pt} % ADDED VSPACE
\penalty\@highpenalty
\endgroup
\fi}
%for mwbk
\def\mw@seccntformat#1{#1\enspace} % para tirar o ponto após número do capítulo
\renewcommand*\chapter@toc{%
\ifHeadingNumbered\typeout{\@chapapp\space\thechapter.}\fi
\addcontentsline{toc}{chapter}{%
\ifHeadingNumbered
\mw@seccntformat{\color{ForestGreen}\@chapapp\ \HeadingNumber}%
\fi
\color{yellow}\hfill\HeadingTOCText\hfill}%
\addtocontents{lof}{\protect\addvspace{10\p@}}%
\addtocontents{lot}{\protect\addvspace{10\p@}}%
}
\makeatother
\begin{document}
\def\manbreak{\hfill\hfill\\\mbox{}\hfill\hfill}
\tableofcontents
\let\manbreak\relax
\chapter{Some chapter}
\blinddocument
\blinddocument
\chapter{This is a really extra long \manbreak chapter name}
\end{document}


While I don't know how to allow for \pars in the definition of \l@chapter or \mw@seccntformat, a similar approach with \manbreak can be used, here redefining \def\mw@seccntformat#1{#1\Manbreak}, where \Manbreak likewise has one definition in the TOC, and is \relaxed afterwards.
\documentclass{mwbk}
\usepackage[utf8]{inputenc}
\usepackage{mathptmx}
\usepackage{graphicx}
\usepackage[dvipsnames]{xcolor}
\usepackage{blindtext}
\makeatletter
\renewcommand*\l@chapter[2]{%
\ifnum \c@tocdepth >\m@ne
\addpenalty{-\@highpenalty}%
\vskip 1.0em \@plus\p@
\setlength\@tempdima{1.5em}%
\begingroup
\parindent \z@ \rightskip \@pnumwidth
\parfillskip -\@pnumwidth
\leavevmode \bfseries \LARGE % added for large font for chapters
\advance\leftskip\@tempdima
\hskip -\leftskip %\@chapapp~ \par % added \@chapapp~ \par
#1\nobreak \raggedright % originally \hfil
%added for leaders (dots) in chapter in toc
%\xleaders\hbox{$\m@th
% \mkern \@dotsep mu\hbox{.}\mkern \@dotsep
% mu$}\hfill%
%%%
\nobreak\hb@xt@\@pnumwidth{\hss \itshape \fbox{#2}}\par\vspace{12pt} % ADDED VSPACE
\penalty\@highpenalty
\endgroup
\fi}
%for mwbk
\def\mw@seccntformat#1{#1\Manbreak} % para tirar o ponto após número do capítulo
\renewcommand*\chapter@toc{%
\ifHeadingNumbered\typeout{\@chapapp\space\thechapter.}\fi
\addcontentsline{toc}{chapter}{%
\ifHeadingNumbered
\mw@seccntformat{\color{ForestGreen}\@chapapp\ \HeadingNumber}%
\fi
\color{yellow}\hfill\HeadingTOCText\hfill}%
\addtocontents{lof}{\protect\addvspace{10\p@}}%
\addtocontents{lot}{\protect\addvspace{10\p@}}%
}
\makeatother
\begin{document}
\def\Manbreak{\\\mbox{}}
\def\manbreak{\hfill\hfill\\\mbox{}\hfill}
\tableofcontents
\let\manbreak\relax
\let\Manbreak\relax
\chapter{Some chapter}
\blinddocument
\blinddocument
\chapter{This is a really extra long \manbreak chapter name}
\end{document}
