\mathbb not working in \underset

The problem is caused by package flexisym. Not only \mathbb{R} is affected but also \sup is in an italic font. Anyway, \underset is not the right command for putting something below the "operator" \sup. This is done by a subscript. Compare:

\documentclass{article}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{flexisym}

\begin{document}
\[
  \underset{x \in \mathbb{R}}{\sup} = \sup_{x \in \mathbb{R}}
\]
\end{document}

If you do not need package flexisym, then do not load it.

If you delete \usepackage{flexisym} it works well.

In my experience, to put so many packages, often it create conflict or bad compilation. It is important, also, the order of the list the various packages.

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{amssymb,amsthm}
\usepackage{enumitem}
\usepackage{microtype}
\usepackage{idxlayout}
\usepackage{lmodern}
\usepackage{float}
\usepackage{amsthm}
%\usepackage{flexisym}
\usepackage{graphicx}
\usepackage{mathtools}

\begin{document}    

$\underset{x \in \mathbb{R}}{\sup}$
\end{document}