The @online entry type does not use the field institution, but it knows the similar field organization (most types know at least one of institution, organization and publisher, some types support publisher and organization). You can find the list of known fields for each entry type in section 2.1 Entry Types of the biblatex manual.
By default organization is printed without special formatting (just like institution and publisher), if you want to print it in italics (emphasised), you can use \DeclareListWrapperFormat if you use the biblatex 3.12 or above.
%\RequirePackage{filecontents}
\begin{filecontents}{\jobname.bib}
@online{musk,
author = {Elon Musk},
title = {Tesla Model X},
url = {https://www.tesla.com/modelx?redirect=no},
date = {2017-07-04},
organization = {Tesla},
urldate = {2018-10-16},
}
\end{filecontents}
\documentclass{article}
\usepackage[style=authortitle,]{biblatex}
\addbibresource{\jobname.bib}
\DeclareListWrapperFormat{organization}{\mkbibemph{#1}}
\begin{document}
Let's cite! \footcite{musk}
\printbibliography
\end{document}
If you are using a biblatex version older than 3.12, where \DeclareListWrapperFormat is not yet available you can use
\DeclareListFormat{organization}{%
\usebibmacro{list:delim}{#1}%
\mkbibemph{#1}\isdot
\usebibmacro{list:andothers}}
In case you are using an exporter that stubbornly insist on using institution instead of organization you can let Biber remap the fields
\DeclareSourcemap{
\maps[datatype=bibtex]{
\map{
\pertype{online}
\pertype{electronic}
\step[fieldsource=institution]
\step[fieldset=organization, origfieldval]
}
}
}
If you don't want to show the organization/institution of the website, and are more interested in the 'overall title' of the website, you may want to try maintitle, but that would have to be patched into the driver first, see How to include the title of the website in bibliography.
