How to ensure that vectors have the same width (equation)?

You may work with \hphantom.

\documentclass{article}
\usepackage{enumitem}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amsthm}
\usepackage{amsfonts} \newcommand\N{\mathbb{N}}
\usepackage[a4paper, total={6in, 8in}]{geometry}
\newcommand{\powerset}{\raisebox{.15\baselineskip}{\Large\ensuremath{\wp}}}
\newcommand\tab[1][1cm]{\hspace*{#1}}
\title{ Groupwork}
\author{some names}
\date{June 2018}
\begin{document}
\maketitle

\section*{Exercise 1}
\begin{enumerate}[label={\alph*)}]
\item
%some text here 
Mathijs's: $\begin{bmatrix}\hphantom{-2}\makebox[0pt][r]{2\;\,} \\ 1 \\ 2 \\ 0 \\ 1 \\ 1 \\ 0 \\ 1 
\end{bmatrix}$, Abdul: $\begin{bmatrix}2 \\ -2 \\ 0 \\ 1 \\ 1 \\ 2 \\ -1 \\ 
0 \end{bmatrix}$, Oleksandr: $\begin{bmatrix}0 \\ 1 \\ 0 \\ -1\\ -2 \\ 0 \\ 
2 \\ 2 \end{bmatrix}$, Laurens: $ \begin{bmatrix}-1 \\ 2 \\ 2 \\ 1 \\ 1 \\ 
-2 \\ 0\\ 0 \end{bmatrix}$

\end{enumerate}
\end{document}

But personally I would not do that.

Use \makebox:

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{enumitem}
\usepackage{amsmath,amsthm}
\usepackage{amsfonts}
\usepackage{calc}
\usepackage{graphicx}
\usepackage[a4paper, total={6in, 8in}]{geometry}

\newcommand{\powerset}{{\mathpalette\myraise\wp}}
\newcommand{\myraise}[2]{%
  \vcenter{\hbox{\scalebox{1.4}{$#1#2$}}}%
}

\newcommand\tab[1][1cm]{\hspace*{#1}}
\newcommand\N{\mathbb{N}}

\newcommand{\adjwd}[2]{\makebox[\widthof{$#1$}]{#2}}

\title{Groupwork}
\author{some names}
\date{June 2018}

\begin{document}
\maketitle

\section*{Exercise 1}
\begin{enumerate}[label={\alph*)}]
\item
%some text here 
Mathijs's:
  $\begin{bmatrix}\adjwd{-2}{2} \\ 1 \\ 2 \\ 0 \\ 1 \\ 1 \\ 0 \\ 1 \end{bmatrix}$,
Abdul:
  $\begin{bmatrix}2 \\ -2 \\ 0 \\ 1 \\ 1 \\ 2 \\ -1 \\ 0 \end{bmatrix}$,
Oleksandr:
  $\begin{bmatrix}0 \\ 1 \\ 0 \\ -1\\ -2 \\ 0 \\ 2 \\ 2 \end{bmatrix}$,
Laurens: $ \begin{bmatrix}-1 \\ 2 \\ 2 \\ 1 \\ 1 \\ -2 \\ 0\\ 0 \end{bmatrix}$

\item Power set $\powerset(X)$
\end{enumerate}

\end{document}

I also suggest a better way to typeset the power set symbol (although I wouldn't use Weierstrass’ p for the purpose).

Another solution, with the small makebox package. In addition, a variant with medium-sized matrices, which look better, in my opinion, for matrices with many rows and not so many columns:

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[a4paper, total={6in, 8in}]{geometry}
\usepackage{enumitem}
\usepackage{mathtools, nccmath}
\usepackage{bigstrut, makebox}
\newenvironment{mbmatrix}[1][c]{\medsize\begin{bmatrix*}[#1]\bigstrut[t]}{\end{bmatrix*}\endmedsize}

\title{ Groupwork}
\author{some names}
\date{June 2018}

\begin{document}
\maketitle

\section*{Exercise 1}
\begin{enumerate}[label={\alph*)}]
\item
%some text here
Mathijs's: $\begin{bmatrix}\makebox*{$-2$}{$ 2 $} \\ 1 \\ 2 \\ 0 \\ 1 \\ 1 \\ 0 \\ 1
\end{bmatrix}$, Abdul: $\begin{bmatrix}2 \\ -2 \\ 0 \\ 1 \\ 1 \\ 2 \\ -1 \\
0 \end{bmatrix}$, Oleksandr: $\begin{bmatrix}0 \\ 1 \\ 0 \\ -1\\ -2 \\ 0 \\
2 \\ 2 \end{bmatrix}$, Laurens: $ \begin{bmatrix}-1 \\ 2 \\ 2 \\ 1 \\ 1 \\
-2 \\ 0\\ 0 \end{bmatrix}$\medskip
%
\item
%%some text here
Mathijs's: $\begin{mbmatrix}\makebox*{$-2$}{$ 2 $} \\ 1 \\ 2 \\ 0 \\ 1 \\ 1 \\ 0 \\ 1
\end{mbmatrix}$, Abdul: $\begin{mbmatrix}2 \\ -2 \\ 0 \\ 1 \\ 1 \\ 2 \\ -1 \\
0 \end{mbmatrix}$, Oleksandr: $\begin{mbmatrix}0 \\ 1 \\ 0 \\ -1\\ -2 \\ 0 \\
2 \\ 2 \end{mbmatrix}$, Laurens: $ \begin{mbmatrix}-1 \\ 2 \\ 2 \\ 1 \\ 1 \\
-2 \\ 0\\ 0 \end{mbmatrix}$
\end{enumerate}

\end{document}