How to import the summation symbol from the "STIX" font

You need to set-up the LS2 encoding and avoid overwriting the largesymbols:

\documentclass{article}

\usepackage{amssymb,amsmath}

\makeatletter
\DeclareFontEncoding{LS2}{}{\noaccents@}
\makeatother

\DeclareFontSubstitution{LS2}{stix}{m}{n}
\DeclareSymbolFont{xlargesymbols}{LS2}{stixex}{m}{n}
\DeclareMathSymbol{\sumop}{\mathop}{xlargesymbols}{"B3}

\begin{document}

Hi, here is the $\sumop$ and
\[ \sumop \]
symbol of STIX and here are the standard $\sum$ and
\[ \sum \]
symbols.
\end{document}

In your sample file the first error in the .log file is

LaTeX Error: Encoding scheme `LS2' unknown.

telling that the encoding scheme has not been defined. Digging around the stix.sty file, leads to how that files sets-up the encoding.

Andrew’s answer is only partial.

If you want to full substitute \sum with the STIX symbol, you have to redefine \sum@:

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

\makeatletter
\DeclareFontEncoding{LS2}{}{\noaccents@} 
\DeclareFontSubstitution{LS2}{stix}{m}{n}

\DeclareSymbolFont{largesymbolsSTIX}{LS2}{stixex}{m}{n}
\DeclareMathSymbol{\sum@}{\mathop}{largesymbolsSTIX}{"B3}
\makeatother

\begin{document}         

Hi, here is the $\sum$ and $\displaystyle\sum$ symbol of STIX

\end{document}

1. The amsmath package saves \sum in \sum@ and then redefines \sum to be a complex macro; it does so in order to comply with the \dots features and to respect the sumlimits or nosumlimits options.

2. Just defining \sumop will not make the symbol comply with the above features.

3. The font encoding LS2 is not known to LaTeX, so it should be declared before referring to a font using it. The required \DeclareFontEncoding command can be found in stix.sty.