Visible Speech font?
There seems to be a MetaFont available, have a look at: https://ctan.org/tex-archive/language/vispeech
You can install that by following http://www.tex.ac.uk/FAQ-instmffont.html
There are two free TrueType fonts for Visible Speech: Herman Miller’s Teamouse VS and Mark Shoulson’s VSMeta Plain (PUA). The latter is by the author of the MetaFont package referred to by TobiBS. Both rely on the Private Use Area, since Visible Speech is not (yet) in Unicode.
Building on Thérèse's answer, here's a XeTeX document that shows the characters in the VSMeta font using XeTeX. The code points are mixed using some easily accessible characters (which seem to be for diacritic characters) but mainly with high private use area points. Thanks to Ulrike Fischer for helping with the high value pgffor
loop. For easy input it would probably make sense to package this with sensible macro names for each glyph, or possibly create a Teckit mapping for it.
\documentclass{article}
\usepackage{pgfmath,pgffor}
\usepackage[margin=1in]{geometry}
\usepackage{xfp}
\usepackage{multicol}
\usepackage{fontspec}
\newfontfamily\visible{VSMeta Plain (PUA)}
\parindent=0pt
\newcommand{\chardisplay}[1]{{\ttfamily#1\ }{\visible\char"#1}\par}
\begin{document}
\begin{multicols}{8}
\chardisplay{00C8}
\chardisplay{00C9}
\chardisplay{00D5}
\chardisplay{00D6}
\chardisplay{00DA}
\chardisplay{00DB}
\chardisplay{0114}
\chardisplay{0116}
\foreach\x in {280,...,295}{
\pgfmathHex{\x}
\chardisplay{\pgfmathresult}
}
\foreach\x in {305,...,308}{
\pgfmathHex{\x}
\chardisplay{\pgfmathresult}
}
\foreach\x in {313,...,315}{
\pgfmathHex{\x}
\chardisplay{\pgfmathresult}
}
% The next set are the PUA glyphs
% code points 59264-59446 (too big for pgffor)
%
\foreach \x in {0,...,182}{
\pgfmathHex{\fpeval{\x+59264}}
\chardisplay{\pgfmathresult}
}
\end{multicols}
\end{document}