I propose two solutions, based on the versions of leqslant
in amssymb
and mathabx
(without loading the latter fonts):
\documentclass{article}
\usepackage{amssymb}
\usepackage{graphicx}
\newcommand{\dbvee}{\mathbin{\rotatebox[origin=c]{-90}{$ \geqslant $}}}
\DeclareFontFamily{U}{matha}{\hyphenchar\font45}
\DeclareFontShape{U}{matha}{m}{n}{ <-6> matha5 <6-7> matha6 <7-8>
matha7 <8-9> matha8 <9-10> matha9 <10-12> matha10 <12-> matha12 }{}
\DeclareSymbolFont{matha}{U}{matha}{m}{n}
\DeclareMathSymbol{\abxgeqslant}{\mathord}{matha}{"A5}
\newcommand{\abxdbvee}{\mathbin{\rotatebox[origin=c]{-90}{$ \abxgeqslant $}}}
\begin{document}
\[ A\dbvee B \qquad A\abxdbvee B\]%
\end{document}

Edit:
To have the same symbol with \bigvee
, you can add this to you preamble:
\usepackage{stackengine, mathtools}
\newcommand{\bigeqvee}{\stackMath\mathop{\stackinset{l}{-2pt}{c}{-0.5pt}{\scalebox{1.35}{$ \setminus $}}{\displaystyle\bigvee}}}
and type in the body of your document
\[ A = \bigeqvee_{\mathclap{i\in I}} A_i \]%
(of course this works only for display style
. For a more general use, one would have to use \mathpalette
, in order to adapt the code to textstyle, script style ans=d scriptscriptcstyle –) but will you need them?

\(\mathbb{V}\)
withamssymb
package loaded works for you?