The double brackets are \lBrack and rBrack in several packages, including unicode-math, stix, stix2 and fdsymbol. A convenient way to use them is in a paired math delimiter from mathtools.
\documentclass{article}
\usepackage{mathtools}
\usepackage{unicode-math}
\DeclarePairedDelimiter\Brackets{\lBrack}{\rBrack}
\begin{document}
[ \Brackets*{\overrightarrow{\mathcal A};}
]
\end{document}

You can manually specify the size of the brackets with, e.g., \Brackets[\big]. Here, I added a bit of space on the right to compensate for the arrow smashing into the right delimiter, which is not a great solution.
You could also adjust the position of the arrow with \skew:
\documentclass{article}
\usepackage{mathtools}
\usepackage{unicode-math}
\DeclarePairedDelimiter\Brackets{\lBrack}{\rBrack}
\newcommand\Aaarow{\skew{-2}\overrightarrow{\mathcal{A}}}
\begin{document}
[ \Brackets*{\Aaarow}
]
\end{document}

Or you could use a font whose \overrightarrow does not extend as far, such as STIX Two Math.
$\left [\!\left[ \overrightarrow{\mathcal A} \right]\!\right]$solve? – SolidMark Jun 24 '21 at 17:41$\left [\!\left[ \overrightarrow{\mathcal A} \right]\!\right]$will separate the double bracket. I want to keep them share the same horizontal edge. – Akira Jun 24 '21 at 17:43[\![and it scales as a single character – David Carlisle Jun 24 '21 at 18:00