\documentclass{amsart}

\usepackage{orcidlink}
\usepackage{lipsum} % To generate some text for the article via \lipsum

\begin{document}

\title{Lorem ipsum dolor sit amet}

% Specify running author explicitly to exclude the ORCID icon
\author[Author One]{Author One\orcid{0000-0000-0000-0000}}
\author[Author Two]{Author Two\orcidnum{0000-0000-0000-0000}}

\begin{abstract}
  \lipsum[1] % Generate some random text
\end{abstract}

\maketitle

\section{Lorem ipsum dolor sit amet}
\lipsum[2-4] % Generate some random text

\end{document}
