How can I center the text inside each cell in a table vertically?

In addition to vertically centering the contents of the table, I have also implemented the following changes in the below MWE:

• Used tabularx in order to make sure, the table fits into the textwidth. In the original example, it was a lot wider than the textwidth. You got an voerfull box warning informing you about that.
• In order to vertically center the contents, I have added renewcommand{tabularxcolumn}[1]{m{#1}}. Originally, the X type column is derived from the p type column and therefore aligns ist contents just as in your original example. The above line now redefindes the X type column to act like a vertically centered m type column. (There is no need to explicitly add the array package here, since tabularx already loads it.)
• I have added the makecell package in order to replace the nested tabulars and the multicolumns in the column headers.
• I have removed the multirow commands in the last table row, as well as the empty row there.
• I have added the enumitem package and defined a new customized tabenum environment for the use in tables. This environment saves you quite a lot of space above, below and on the left.

documentclass{article}
%usepackage{multirow} % no longer needed
usepackage{graphicx}

usepackage{makecell}
renewcommand{theadfont}{normalsize}

usepackage{tabularx}
renewcommand{tabularxcolumn}[1]{m{#1}}

usepackage{enumitem}
newlist{tabitem}{enumerate}{1}

setlist[tabitem,1]{label*=textbullet,
                    leftmargin=*,
                    nosep,
                    leftmargin=*,
                    before=begin{minipage}[t]{hsize},
                    after=end{minipage}}

begin{document}
begin{table}[h]
  centering
  begin{tabularx}{textwidth}{|c|c|X|}
    hline
    thead{Position} 
      & thead{Person(s)} 
        & thead{Role} 
    hline
    Co-head 
      & makecell{Islam Mahdy Youssef Amr} 
        & begin{tabitem}
            item Reviewing the content of each session previously.  
            item Reviewing the progression of the academic sessions whether they for the Curriculum sessions or the standardized tests' sessions.
            item Having continuous check on the magazine to ensure the conformity of the topics discussed to the club's goal  
          end{tabitem}     
    hline
    Vice head 
      & Ahmed Magdy 
        & loremipsum  
    hline
    Head of Academics 
      & Mohamed Badr 
        &  begin{tabitem}
             item Preparing the content of each session.  
             item Managing the time table of the weekly-held sessions.
             item Conducting mini quizzes each week to make sure the students have comprehended the previous session.   
           end{tabitem}            
    hline
    Magazine 
      &makecell{Ali Ashraf  Ahmed Hamed} 
        & sdsdsdsds 
    hline
  end{tabularx}
end{table}


end{document}

Here are some simplifications and improvements for the vertical spacing, in particular for the lists, loading the enumitem and makecell packages.

documentclass{article}
usepackage{array, multirow, makecell} 
usepackage{enumitem}

begin{document}

begin{table}[h]
    centeringsetlength{extrarowheight}{2pt}
    setlist[itemize]{wide = 0pt, leftmargin = *, topsep =3pt, after =vspace*{dimexprtopsep-baselineskip}}
    begin{tabular}{|c|c| m{3in}|}
        hline
        Position & multicolumn{1}{c|}{Person(s)} & multicolumn{1}{c|}{Role}  hline
        Co-head &
        begin{tabular}[c]{@{}c@{}}Islam Mahdy Youssef Amrend{tabular} &
        begin{itemize}
            item Reviewing the content of each session previously.
            item Reviewing the progression of the academic sessions whether they for the Curriculum sessions or the standardized tests' sessions.
            item Having continuous check on the magazine to ensure the conformity of the topics discussed to the club's goal
        end{itemize}  hline
        Vice head &
        Ahmed Magdy &
    loremipsum  hline
        Head of Academics & Mohamed Badr & begin{itemize}
            item Preparing the content of each session.
            item Managing the time table of the weekly-held sessions.
            item Conducting mini quizzes each week to make sure the students have comprehended the previous session.
        end{itemize}  hline
        multirow{2}{*}{Magazine} &
        multirowcell{2}{Ali Ashraf  Ahmed Hamed} &
        multirow{2}{*}{sdsdsdsds} 
        & &  hline
    end{tabular}
end{table}

end{document}