Creating a variable size math operator

I wouldn't use a full \circ, because it's too big. If we scale it by half and use the bold version to compensate, the symbol can accommodate high subscripts without overlapping.

\documentclass{article}
\usepackage{amsmath,graphicx}

\makeatletter
\newcommand{\bigveecirc}{\DOTSB\bigveecirc@\slimits@}
\newcommand{\bigveecirc@}{\mathop{\mathpalette\bigveecirc@@\relax}}
\newcommand{\bigveecirc@@}[2]{% #1 is a style selection, #2 is unused
  \vcenter{\vbox{\halign{\hfil##\hfil\cr
    $\m@th#1\bigvee$\cr
    \noalign{\nointerlineskip}
    \raisebox{\dimexpr-\height\ifx#1\displaystyle+0.3ex\fi}[0pt][0pt]{%
      \scalebox{0.5}{$\m@th#1\boldsymbol{\circ}$}%
    }\cr
  }}}%
}
\makeatother

\begin{document}

\[
\bigveecirc_{B}\bigvee_{n=1}^\infty\bigveecirc_{n=1}^\infty a_n
\]
\begin{center}
$
\textstyle\bigvee_{n=1}^\infty\bigveecirc_{n=1}^\infty a_n
\quad
\scriptstyle\bigvee_{n=1}^\infty\bigveecirc_{n=1}^\infty a_n
\quad
\scriptscriptstyle\bigvee_{n=1}^\infty\bigveecirc_{n=1}^\infty a_n
$
\end{center}

\end{document}

Another alternatives to have a symbol for a math operator that looks like $\vee$, with a $\circ$ attached at the bottom vertex. You can see also the bit differences of height of the symbol $\circ$.

\documentclass[a4paper,12pt]{article}
\usepackage{mathtools,amssymb}
\begin{document}
Sequences of codes:
\[\underset{\circ}{\smash\bigvee}, \underset{\circ}{\bigvee}\]
\[\bigvee_{\substack{\circ}}, 
\underset{\circ}{\smash\prod}\]
\[\underset{\circ}{\prod}, 
\prod_{\substack{\circ}}\]
\[\sum_{\substack{\circ}}, 
\underset{\circ}{\smash\sum}, \underset{\circ}{\sum}\]

\end{document}

The \bigvee directive may be what you're looking for.

\documentclass{article}
\begin{document}
$\vee \quad 
 \bigvee\limits_{\circ} \quad 
 \sum\limits_{i} \quad
 \displaystyle
 \bigvee_{\circ} \quad 
 \sum_{i}$
\end{document}