How can I change the build system of python.sty from Python2 to Python3?

The call to python is hard-coded in the package. You can make it to call python3 with a patch:

\documentclass{article}
\usepackage{python}
\usepackage{etoolbox}

\patchcmd{\endpython}{python }{python3 }{}{}

\begin{document}
\begin{python}
import sys
print (sys.version)
\end{python}
\end{document}

In the log file we find

runsystem(cat  jay.py | python3 > jay.py.out 2> jay.py.err)...executed.

and this is the PDF output

An extended version where you can switch from one version to the other; the \setpython command obeys the normal scoping rules.

\documentclass{article}
\usepackage{python}
\usepackage{etoolbox}

\makeatletter
\patchcmd{\endpython}{python }{python\python@version\space}{}{}
\newcommand{\setpython}[1]{%
  \if3#1\relax
    \def\python@version{3}%
  \else
    \def\python@version{}%
  \fi
}
\makeatletter
\setpython{2} % initialize

\begin{document}
\begin{python}
import sys
print (sys.version)
\end{python}

\bigskip

\setpython{3}

\begin{python}
import sys
print (sys.version)
\end{python}

\end{document}