Space above signature line

Use an invisible object with the desired height.

I also added \par instead of \\ that's not suitable for the job and \hfill instead of \hspace{2cm}.

\documentclass[a4paper]{article}
\usepackage{tabularx}

\newcommand{\signatureline}[1]{%
  \par\noindent
  \begin{tabular}{@{}p{6cm}@{}}
    \rule{0pt}{3cm}
    \\ \hline \\[-.75\normalbaselineskip]
    #1
  \end{tabular}\hfill
  \begin{tabular}{@{}p{4cm}@{}}
    \rule{0pt}{3cm}
    \\ \hline \\[-.75\normalbaselineskip]
    Date
  \end{tabular}\par
}

\newcommand{\approval}{%
  \par\vspace*{\fill} % Puts signature lines in the same spot of each page for easy stamping
  \section*{Approvals}
  \signatureline{Operator} % Putting \vspace after this doesn't put space between these signaturelines
  \signatureline{Authority}
  \clearpage
}

\begin{document}
Some text
\approval
\end{document}

Another solution is to add the same you did to the hline. \\[3cm]

\documentclass[a4paper]{article}
\usepackage{tabularx}

\newcommand{\signatureline}[1]{%
  \begin{tabular}{@{}p{6cm}@{}}
   % \vspace{3cm} % Increasing this seems to have no effect
    \\[3cm]
     \hline \\[-.75\normalbaselineskip]
    #1
  \end{tabular}
  \hspace{2cm}
  \begin{tabular}{@{}p{4cm}@{}}
   % \vspace{3cm} % Increasing this seems to have no effect
    \\[3cm]
     \hline \\[-.75\normalbaselineskip]
    Date
  \end{tabular}
  \\
}

\newcommand{\approval}{%
  \par\vspace*{\fill} % Puts signature lines in the same spot of each page for easy stamping
  \section*{Approvals}
  \signatureline{Operator} % Putting \vspace after this doesn't put space between these signaturelines
  \signatureline{Authority}
  \clearpage
}

\begin{document}
Some text
\approval
\end{document}