I don’t know precisely what the journal you want to submit to accepts. but here’s a pared-down version of the template that adds this Cyrillic character, in both text and math mode. Since the template states that you should use the libertine font package, it takes the Cyrillic alphabet from Libertinus Serif and Libertinus Sans.
%%%% Generic manuscript mode
\documentclass[manuscript,screuen,review]{acmart}
\usepackage{iftex}
\ifPDFTeX
\usepackage[T2A,T1]{fontenc}
\usepackage{newunicodechar, substitutefont}
\substitutefont{T2A}{LinuxLibertineT-TLF}{LibertinusSerif-TLF}
\substitutefont{T2A}{LinuxBiolinumT-TLF}{LibertinusSans-TLF}
\newunicodechar{И}{{\fontencoding{T2A}\selectfont\symbol{"C8}}}
\fi
\newcommand\mitcyrI{\mathord{\text{\rmfamily\mdseries\itshape И}}}
\newcommand\mupcyrI{\mathord{\text{\rmfamily\mdseries\upshape И}}}
\newcommand\mbfitcyrI{\mathord{\text{\rmfamily\bfseries\itshape И}}}
\newcommand\mbfupcyrI{\mathord{\text{\rmfamily\bfseries\upshape И}}}
\newcommand\msfupcyrI{\mathord{\text{\sffamily\mdseries\upshape И}}}
%%
%% \BibTeX command to typeset BibTeX logo in the docs
\AtBeginDocument{%
\providecommand\BibTeX{{%
\normalfont B\kern-0.5em{\scshape i\kern-0.25em b}\kern-0.8em\TeX}}}
%% Rights management information. This information is sent to you
%% when you complete the rights form. These commands have SAMPLE
%% values in them; it is your responsibility as an author to replace
%% the commands and values with those provided to you when you
%% complete the rights form.
\setcopyright{acmcopyright}
\copyrightyear{2018}
\acmYear{2018}
\acmDOI{10.1145/1122445.1122456}
%%
%% end of the preamble, start of the body of the document source.
\begin{document}
%%
%% The "title" command has an optional parameter,
%% allowing the author to define a "short title" to be used in page headers.
\title{The Name of the Title is Hope}
\maketitle
\section{Introduction}
The Cyrillic letter И can mean the variable \(\mitcyrI\), the constant \(\mupcyrI\), the vector \(\mbfitcyrI\) or the constant vector \(\mbfupcyrI\). Other forms such as \textsf{И} for tensors could be defined similarly as \(\msfupcyrI\).
\end{document}
\endinput
%%
%% End of file `sample-manuscript.tex'.

An alternative would be to \DeclareMathSymbol, although this would require you to use up one of the limited number of math alphabets you are allowed to define in legacy TeX.
Edit
You mentioned in a comment compiling this in XeLaTeX. The template does not seem to have been set up to support that, but this quick and dirty hack appears to work. Check with the journal.
\RequirePackage[no-math]{fontspec}
\defaultfontfeatures[LinLibertine]{NFSSFamily=LinuxLibertineT-TLF}
\documentclass[manuscript,screuen,review]{acmart}
\usepackage[nomath]{libertinus}
\newcommand\mitcyrI{\mathord{\text{\rmfamily\mdseries\itshape И}}}
\newcommand\mupcyrI{\mathord{\text{\rmfamily\mdseries\upshape И}}}
\newcommand\mbfitcyrI{\mathord{\text{\rmfamily\bfseries\itshape И}}}
\newcommand\mbfupcyrI{\mathord{\text{\rmfamily\bfseries\upshape И}}}
\newcommand\msfupcyrI{\mathord{\text{\sffamily\mdseries\upshape И}}}
%%
%% \BibTeX command to typeset BibTeX logo in the docs
\AtBeginDocument{%
\providecommand\BibTeX{{%
\normalfont B\kern-0.5em{\scshape i\kern-0.25em b}\kern-0.8em\TeX}}}
%% Rights management information. This information is sent to you
%% when you complete the rights form. These commands have SAMPLE
%% values in them; it is your responsibility as an author to replace
%% the commands and values with those provided to you when you
%% complete the rights form.
\setcopyright{acmcopyright}
\copyrightyear{2018}
\acmYear{2018}
\acmDOI{10.1145/1122445.1122456}
%%
%% end of the preamble, start of the body of the document source.
\begin{document}
%%
%% The "title" command has an optional parameter,
%% allowing the author to define a "short title" to be used in page headers.
\title{The Name of the Title is Hope}
\maketitle
\section{Introduction}
The Cyrillic letter И can mean the variable \(\mitcyrI\), the constant \(\mupcyrI\), the vector \(\mbfitcyrI\) or the constant vector \(\mbfupcyrI\). Other forms such as \textsf{И} for tensors could be defined similarly as \(\msfupcyrI\).
\end{document}
\usepackage[T2A,T1]{fontenc}. This sets T1 as the default encoding and T2A as the supplemental one, instead of the other way around. – Davislor May 03 '20 at 10:05acmartdocument in XeLaTeX and just use Unicode. – Davislor May 03 '20 at 10:11