Your MWE has bunch of errors, the most of them I succeeded to eliminate, so the MWE is now at least possible to compile ...
Edit (1):
Meanwhile I dug through pgfplotstable manual and SE in searching of not resolved issues in my answer. This was quite fruitful; I was able to found my mistakes, which I introduced when correcting errors in your MWE ... The resulting new code and image of obtained table are given below.
\documentclass[margin=5mm,preview]{standalone}
\usepackage{siunitx} % Formats the units and values
\sisetup{ % setup siunitx ...
round-mode = places, % rounds numbers
round-precision = 2, % to 3 places
per-mode = symbol, % kg/dm^3 instead kgm^{-3}
group-four-digits = true, % for 1 234,567
}
\usepackage{booktabs} % for table rules
\usepackage{pgfplotstable} % Generates table from .csv
\usepackage{filecontents} % <--- important: enable table
% refreshing at each compilation
\usepackage[hang,bf,small]{caption}
%---------------------------------------------------------------%
\begin{filecontents*}{mytable.csv}
Chem.; Avg. Conc.; Avg. Conc. Norm.; Conc. unit; Mass sum; Mass unit
Ammonium ; 159083.33; 114450.21; \si{\micro\gram\per\liter}; 2839.463; \si{\kilo\gram}
Ammonium* ; 1234.123; 4567.890; \si{\micro\gram\per\liter}; 2839.46; \si{\kilo\gram}
\end{filecontents*}
%---------------------------------------------------------------%
\begin{document}
Test of use \verb+siunitx+ units syntax in text \si{\micro\gram\per\liter} and \si{\kilo\gram},
\captionof{table}{Some caption text}
\pgfplotstabletypeset[
multicolumn names,
col sep=semicolon, % the separator in our .csv file
string type, % added in hopes of enabling alphabetic input.
header=has colnames,
every head row/.style={before row=\toprule,after row=\midrule},
every last row/.style={after row=\bottomrule},
display columns/0/.style={string type},
display columns/1/.style={column type={S[table-format=7.3]}},% use
display columns/2/.style={column type={S[table-format=7.3]}},% siunitx
display columns/3/.style={string type}, % units as string
display columns/4/.style={column type={S[table-format=5.3]}},% for formating
display columns/5/.style={string type}, % units as string
]{mytable.csv}
\end{document}
As you can see, in table I introduced S column type of column and used siunitx engine for number formatting in columns.
Warning: Due to lengthy column heads, the table is wider than usual \textwidth in article document-class.

Edit (2):
Meanwhile, Stefan Pinnow in his comment described alternative solution (which for unknown reason didn't work before, now I cannot remember, what was wrong :-( ). However, corrected code is given below.
\documentclass[margin=5mm,preview]{standalone}
\usepackage{siunitx} % Formats the units and values
\sisetup{ % setup siunitx ...
round-mode = places, % rounds numbers
round-precision = 2, % to 3 places
per-mode = symbol, % kg/dm^3 instead kgm^{-3}
group-four-digits = true, % for 1 234,567
}
\usepackage{booktabs} % for table rules
\usepackage{pgfplotstable} % Generates table from .csv
\usepackage{filecontents} % <--- important: enable table
% refreshing at each compilation
\usepackage[hang,bf,small]{caption}
%---------------------------------------------------------------%
\begin{filecontents*}{mytable.csv}
Chem.; Avg. Conc.; Avg. Conc. Norm.; Conc. unit; Mass sum; Mass unit
Ammonium ; 159083.33; 114450.21; \micro\gram\per\liter; 2839.463; \kilo\gram
Ammonium* ; 1234.123; 4567.890; \micro\gram\per\liter; 2839.46; \kilo\gram
\end{filecontents*}
%---------------------------------------------------------------%
\begin{document}
Test of use \verb+siunitx+ units syntax in text \si{\micro\gram\per\liter} and \si{\kilo\gram},
\captionof{table}{Some caption text}
\pgfplotstabletypeset[
multicolumn names,
col sep=semicolon, % the separator in our .csv file
string type, % added in hopes of enabling alphabetic input.
header=has colnames,
every head row/.style={before row=\toprule,after row=\midrule},
every last row/.style={after row=\bottomrule},
display columns/0/.style={string type},
display columns/1/.style={column type={S[table-format=7.3]}},% use
display columns/2/.style={column type={S[table-format=7.3]}},% siunitx
display columns/3/.style={column type={s}}, % for units
display columns/4/.style={column type={S[table-format=5.3]}},% for formating
display columns/5/.style={column type={s}}, % for units
]{mytable.csv}
\end{document}
Result is the same as before!
scolumn fromsiunitxwith pgfplotstable? – Holene Jan 27 '16 at 13:51siunitxcolumn type isS(nots), and it enables different number formatting. This formatting is on someway integrated inpgfplotstable, so there is seems no need for this column type (you can see this, if you comment\sisetup{...}, the formating of numbers will change). I still learning this awesome (and sometimes seems to be complicated) package ... – Zarko Jan 27 '16 at 14:00Scolumn type inpgfplotstable. – Zarko Jan 27 '16 at 19:34scolumn type: This is available insiunitxand is used to typeset units like the\sicommand. So by replacingstring typebycolumn type={s}in your above code and removing the\si{}and just letting the content of it in the table (e.g.\si{\kilo\gram}by\kilo\gram) also works fine. – Stefan Pinnow Jan 30 '16 at 13:07Invalid numerical input '.'orInvalid numerical input 'e'and similar) – Mar 14 '18 at 08:03