How to Obtain a Random Non-Zero Integer?

Choose a random number between 1 and X (thereby avoiding 0) and multiply it with 1 or -1 based on a uniform distribution (thereby obtaining a sign):

\documentclass{article}

\usepackage{tikz}
\pgfmathsetseed{\number\pdfrandomseed}

\newcommand{\InitVariables}{%
  \pgfmathsetmacro{\a}{int(ifthenelse(rand > 0, 1, -1)*random(1,5))}%
}

\begin{document}

\foreach \x in {1,...,100}{\InitVariables$\a$ }

\end{document}

The first calculation only chooses the number of random values that are needed. The inclusive range [-5, 5] without 0 has 10 values.

\pgfmathsetmacro{\a}{random(0,9)-5}% range [-5, 4]

Then the negative values are in the correct range. The non-negative values are increased by one to move the range [0, 4] to [1, 5].

\pgfmathsetmacro{\a}{int(ifthenelse(\a<0, \a, \a + 1)}%

Full example:

\documentclass{article}
\usepackage{pgf}
\usepackage{pgffor}
\pgfmathsetseed{\number\pdfrandomseed}

\newcommand{\InitVariables}{%
  \pgfmathsetmacro{\a}{random(0,9)-5}%
  \pgfmathsetmacro{\a}{int(ifthenelse(\a<0, \a, \a + 1)}%
}

\begin{document}

\noindent
\foreach \i in {0, ..., 200} {%
  \InitVariables
  \a\space
}

\end{document}

Version with the minimum and maximum values as macros:

\documentclass{article}

\usepackage{pgf}
\usepackage{pgffor}
\pgfmathsetseed{\number\pdfrandomseed}

\newcommand*{\RandomMinimum}{-300}
\newcommand*{\RandomMaximum}{700}

\newcommand*{\InitVariables}{%
  \pgfmathsetmacro{\a}{%
     \RandomMinimum + random(0, int(\RandomMaximum - int(\RandomMinimum)))
  }%
  \pgfmathtruncatemacro{\a}{ifthenelse(\a<0, \a, \a + 1}%
}

\begin{document}

\noindent
\foreach \i in {0, ..., 200} {%
  \InitVariables
  \a\space
}

\end{document}

Generate a number between 0 and 2​x –1, then normalize it: if it is less than x, subtract x, otherwise subtract x and add 1.

\documentclass{article}

\newcommand{\randomdef}[2]{%
  \edef#1{%
    \expandafter\randomdefnormalize\pdfuniformdeviate\numexpr#2*2\relax\foo{#2}%
  }%
}
\def\randomdefnormalize#1\foo#2{%
  \ifnum#1<#2
    \the\numexpr#1-#2\relax
  \else
    \the\numexpr#1-#2+1\relax
  \fi
}

\begin{document}

\randomdef{\arandom}{5}$\arandom$,\space\randomdef{\arandom}{5}$\arandom$,\space
\randomdef{\arandom}{5}$\arandom$,\space\randomdef{\arandom}{5}$\arandom$,\space
\randomdef{\arandom}{5}$\arandom$,\space\randomdef{\arandom}{5}$\arandom$,\space
\randomdef{\arandom}{5}$\arandom$,\space\randomdef{\arandom}{5}$\arandom$,\space
\randomdef{\arandom}{5}$\arandom$,\space\randomdef{\arandom}{5}$\arandom$,\space
\randomdef{\arandom}{5}$\arandom$,\space\randomdef{\arandom}{5}$\arandom$,\space
\randomdef{\arandom}{5}$\arandom$,\space\randomdef{\arandom}{5}$\arandom$,\space
\randomdef{\arandom}{5}$\arandom$,\space\randomdef{\arandom}{5}$\arandom$

\end{document}

The same can of course be done with PGF features.

With xparse and expl3; also an interface to expandably get the number in the required interval. Here I use a different strategy: the random number is generated in the interval –x, x –1; if it's positive, I add 1.

\documentclass{article}
\usepackage{xparse}

\ExplSyntaxOn
\NewDocumentCommand{\randomdef}{mm}
 {
  \wcla_random_def:Nn #1 { #2 }
 }
\DeclareExpandableDocumentCommand{\randomget}{m}
 {
  \wcla_random_get:n { #1 }
 }
\cs_new_protected:Nn \wcla_random_def:Nn
 {
  \cs_set:Npx #1 { \wcla_random_get:n { #2 } }
 }
\cs_new:Nn \wcla_random_get:n
 {
  \__wcla_random_get:f { \fp_eval:n { randint(-#1,#1-1) } }
 }
\cs_new:Nn \__wcla_random_get:n
 {
  \int_compare:nTF { #1 < 0 } { #1 } { \fp_eval:n { #1+1 } }
 }
\cs_generate_variant:Nn \__wcla_random_get:n { f }
\ExplSyntaxOff

\begin{document}

\randomdef{\arandom}{5}$\arandom$,\space\randomdef{\arandom}{5}$\arandom$,\space
\randomdef{\arandom}{5}$\arandom$,\space\randomdef{\arandom}{5}$\arandom$,\space
\randomdef{\arandom}{5}$\arandom$,\space\randomdef{\arandom}{5}$\arandom$,\space
\randomdef{\arandom}{5}$\arandom$,\space\randomdef{\arandom}{5}$\arandom$,\space
\randomdef{\arandom}{5}$\arandom$,\space\randomdef{\arandom}{5}$\arandom$,\space
\randomdef{\arandom}{5}$\arandom$,\space\randomdef{\arandom}{5}$\arandom$,\space
\randomdef{\arandom}{5}$\arandom$,\space\randomdef{\arandom}{5}$\arandom$,\space
\randomdef{\arandom}{5}$\arandom$,\space\randomdef{\arandom}{5}$\arandom$

$\randomget{5}$, $\randomget{5}$, $\randomget{5}$, $\randomget{5}$,
$\randomget{5}$, $\randomget{5}$, $\randomget{5}$, $\randomget{5}$,
$\randomget{5}$, $\randomget{5}$, $\randomget{5}$, $\randomget{5}$,
$\randomget{5}$, $\randomget{5}$, $\randomget{5}$, $\randomget{5}$

\end{document}