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I am using this working example:

\documentclass[a0paper,landscape,final]{baposter}

\usepackage{times}
\usepackage{calc}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{relsize}
\usepackage{multirow}
\usepackage{bm}
\usepackage{tasks}
\usepackage{graphicx}
\usepackage{multicol}
\usepackage{subcaption}
\usepackage{float}
\usepackage{amsmath}
\usepackage[version=4]{mhchem}
\usepackage{pgfbaselayers}
\pgfdeclarelayer{background}
\pgfdeclarelayer{foreground}
\pgfsetlayers{background,main,foreground}
\usepackage{helvet}
%\usepackage{bookman}
\usepackage{palatino}
%
%\usepackage{shortlst}


\usepackage{lipsum}

\usepackage[inline]{enumitem}   
\makeatletter
% This command ignores the optional argument for itemize and enumerate lists
\newcommand{\inlineitem}[1][]{%
\ifnum\enit@type=\tw@
    {\descriptionlabel{#1}}
  \hspace{\labelsep}%
\else
  \ifnum\enit@type=\z@
       \refstepcounter{\@listctr}\fi
    \quad\@itemlabel\hspace{\labelsep}%
\fi}


\usepackage{graphicx}
\graphicspath{ {images/} }


\usepackage[font=small,labelfont=bf]{caption}

\selectcolormodel{cmyk}



\graphicspath{{images/}}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Some math symbols used in the text
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Format 
\newcommand{\Matrix}[1]{\begin{bmatrix} #1 \end{bmatrix}}
\newcommand{\Vector}[1]{\Matrix{#1}}
\newcommand*{\SET}[1]  {\ensuremath{\mathcal{#1}}}
\newcommand*{\MAT}[1]  {\ensuremath{\mathbf{#1}}}
\newcommand*{\VEC}[1]  {\ensuremath{\bm{#1}}}
\newcommand*{\CONST}[1]{\ensuremath{\mathit{#1}}}
\newcommand*{\norm}[1]{\mathopen\| #1 \mathclose\|}% use instead of $\|x\|$
\newcommand*{\abs}[1]{\mathopen| #1 \mathclose|}% use instead of $\|x\|$
\newcommand*{\absLR}[1]{\left| #1 \right|}% use instead of $\|x\|$

\def\norm#1{\mathopen\| #1 \mathclose\|}% use instead of $\|x\|$
\newcommand{\normLR}[1]{\left\| #1 \right\|}% use instead of $\|x\|$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Multicol Settings
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setlength{\columnsep}{0.7em}
\setlength{\columnseprule}{0mm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Save space in lists. Use this after the opening of the list
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\compresslist}{%
\setlength{\itemsep}{1pt}%
\setlength{\parskip}{0pt}%
\setlength{\parsep}{0pt}%
}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Begin of Document
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Here starts the poster
%%%---------------------------------------------------------------------------
%%% Format it to your taste with the options
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\typeout{Poster Starts}
\background{
  \begin{tikzpicture}[remember picture,overlay]%
    \draw (current page.north west)+(-2em,-0em) node[anchor=north west] {\hspace{-2em}\includegraphics[height=1.1\textheight]{silhouettes_background}};
  \end{tikzpicture}%
}
\definecolor{silver}{cmyk}{0,0,0,0.3}
\definecolor{yellow}{cmyk}{0,0,0.9,0.0}
\definecolor{reddishyellow}{cmyk}{0,0.22,1.0,0.0}
\definecolor{black}{cmyk}{0,0,0.0,1.0}
\definecolor{darkYellow}{cmyk}{0,0,1.0,0.5}
\definecolor{darkSilver}{cmyk}{0,0,0,0.1}

\definecolor{lightyellow}{cmyk}{0,0,0.3,0.0}
\definecolor{lighteryellow}{cmyk}{0,0,0.1,0.0}
\definecolor{lighteryellow}{cmyk}{0,0,0.1,0.0}
\definecolor{lightestyellow}{cmyk}{0,0,0.05,0.0}
\definecolor{white}{cmyk}{0,0,0,0}
\begin{poster}{
  % Show grid to help with alignment
  grid=false,
  % Column spacing
  colspacing=1em,
  % Color style
  bgColorOne=lighteryellow,
  bgColorTwo=lighteryellow, %lightestyellow,
  borderColor=reddishyellow,
  headerColorOne=yellow,
  headerColorTwo=reddishyellow,
  headerFontColor=black,
  boxColorOne=lightyellow,
  boxColorTwo=lighteryellow,
  % Format of textbox
  textborder=roundedleft,
  % Format of text header
  eyecatcher=false,
  headerborder=open,
  headerheight=0.08\textheight,
  headershape=roundedright,
  headershade=plain,
  headerfont=\Large\textsf, %Sans Serif
  boxshade=plain,
%  background=shade-tb,
  background=plain,
  linewidth=2pt
  }
  % Eye Catcher
  {} % No eye catcher for this poster. If an eye catcher is present, the title is centered between eye-catcher and logo.
  % Title
  {\sf %Sans Serif
  %\bf% Serif
  Stochastic Modelling with Applications to Michaelis-Menten Kinetics}
  % Authors
  {\sf %Sans Serif
  % Serif
  Daniel McElroy\hspace{3em}
 Supervisor: Dr. Ian Vernon\hspace{3em}
  University of Durham
  }
  % University logo
  {{\begin{minipage}{16em}
    \hfill
    \includegraphics[height=5.5em]{C:/Users/Acer/Desktop/PosterGraphics/durham-university}
  \end{minipage}}
  }

  \tikzstyle{light shaded}=[top color=baposterBGtwo!30!white,bottom color=baposterBGone!30!white,shading=axis,shading angle=30]

  % Width of left inset image
     \newlength{\leftimgwidth}
     \setlength{\leftimgwidth}{0.78em+8.0em}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \headerbox{Bayesian Inference}{name=results,column=0,span=4,row=0, boxColorOne=white!30}{
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{multicols}{2}
[ 
Given our observed data we can use Bayesian inference to deduce our rate constants.We assume our \newline observed data is complete, meaning we know the time and the type of every reaction occuring.
]
\setlength{\columnseprule}{1pt}
\begin{itemize}
\itemsep0em
 \item We call this our sample path, denoted as:
\end{itemize}
\begin{gather*}
 \mathbf x = \{ x(t)\colon t \in [0,T]\}
\end{gather*}
{\par
\begin{itemize}
\itemsep0em
\item $r_j$ signifies the number of times reaction type $j$ occurs in our sample path.
\item As our system is a mass-action kinetic system: 
\end{itemize}
\begin{gather*}
h_j(x,c_j) = c_jg_j(x)
\end{gather*}
\begin{itemize} 
\item The form of the Gillespie algorithm leads us to a likelihood of:
\end{itemize}
\begin{gather*}
L_j(c_j;\mathbf{x}) = c_j^{r_j}\exp\{-c_j\int_0^T g_j(x(t))dt\}, \quad j=1,...,v
\end{gather*}
\par}
{\par
\begin{itemize}
\item We use independent gamma priors for each rate constant, $c_i$.
\end{itemize}
\begin{gather*}
c_j \sim Ga(a_j,b_j), \quad j=1,..,v
\end{gather*}
\begin{itemize}
\item Using Bayes Theorem we are able to produce an updated posterior for our rate constants:
\end{itemize}
\begin{gather*}
c_j|\mathbf {x} \sim Ga(a_j+r_j,b_j+\int_0^Tg_j(x(t))dt), \quad j=1,...,v
\end{gather*}
\par}
\end{multicols}
}
\end{poster}%
%
\end{document}

And I want to separate my columns with a small black vertical line. I know usually I would use the command \setlength{\columnseprule}{1pt} but this doesn't seem to work as I am in a box in my poster. I was looking for a way around this.

1

The line between columns will only show if there is actually more than one column. In your case, except for some contents of the box in question, nothing is type set in multiple columns, hence, no column separator line will appear anywhere else.

So, you can safely put \setlength{\columnseprule}{1pt} in your general settings in the preamble.

In general \columnseprule should always be reset before calling the multicols environment (see: How to change spacing between columns for just one page in the document). So, in case you really only want this one box to use \setlength{\columnseprule}{1pt}, you have to write

...
\setlength{\columnseprule}{1pt}

\begin{multicols}{2}
[ 
Given our observed data we can use Bayesian inference to deduce our rate constants. We assume our \newline observed data is complete, meaning we know the time and the type of every reaction occuring.
]
\begin{itemize}
...

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