Internally acronym uses description. So you can either (1) redefine acronym using your own list structure, or (2) just before acronym redefine description. Here is the second approach. I've added \itemsep0pt\parsep0pt to the standard definition of description.
\documentclass{article}
\usepackage{acronym}
\pagestyle{empty}
\begin{document}
\renewenvironment{description}
{\list{}{\labelwidth0pt\itemindent-\leftmargin
\parsep0pt\itemsep0pt\let\makelabel\descriptionlabel}}
{\endlist}
\begin{acronym}
\acro{JSON}{JavaScript Object Notation}
\acro{JSONP}{JavaScript Object Notation with Padding}
\acro{REST}{Representational State Transfer}
\end{acronym}
\end{document}

UPDATE When perpage option is used, the package adds an extra \\, probably a bug. So in this case we need to redefine the way package prints the items as well:
\documentclass{article}
\usepackage[printonlyused, withpage]{acronym}
\pagestyle{empty}
\makeatletter
\def\AC@@acro#1[#2]#3{%
\ifAC@nolist%
\else%
\ifAC@printonlyused%
\expandafter\ifx\csname acused@#1\endcsname\AC@used%
\item[\protect\AC@hypertarget{#1}{\aclabelfont{#2}}] #3%
\ifAC@withpage%
\expandafter\ifx\csname r@acro:#1\endcsname\relax%
\PackageInfo{acronym}{%
Acronym #1 used in text but not spelled out in
full in text}%
\else%
\dotfill\pageref{acro:#1}% Sputious \\ deleted
\fi
\fi%
\fi%
\else%
\item[\protect\AC@hypertarget{#1}{\aclabelfont{#2}}] #3%
\fi%
\fi%
\begingroup
\def\acroextra##1{}%
\@bsphack
\protected@write\@auxout{}%
{\string\newacro{#1}[\string\AC@hyperlink{#1}{#2}]{#3}}%
\@esphack
\endgroup}
\makeatother
\begin{document}
We use \ac{JSON}, \ac{JSONP}, \ac{REST}.
\renewenvironment{description}
{\list{}{\labelwidth0pt\itemindent-\leftmargin
\parsep0pt\itemsep0pt\let\makelabel\descriptionlabel}}
{\endlist}
\begin{acronym}
\acro{JSON}{JavaScript Object Notation}
\acro{JSONP}{JavaScript Object Notation with Padding}
\acro{REST}{Representational State Transfer}
\end{acronym}
\end{document}
\documentclass{beamer}
\usepackage{graphicx}
\begin{document}
\begin{figure}
\begin{center}
\input{tmp1.tex}
\caption{Enter caption here}
\label{Enter label here}
\end{center}
\end{figure}
\end{document}

\itemsepdid not work but you need the[JSONP]-parameter as well. I did not understand why, but people following this solution should watch for that. – tjati Apr 15 '17 at 10:21[JSONP]) is supposed to be the longest acronym used, as per the acronym pdf manual. It ensures they are aligned. – Clashsoft Apr 25 '20 at 14:03