I wondering there is no option for this as old templates support this on Wikimedia like Template:Clickable_button, I see this because someone used some unicode signs which are not really supported by many systems: 🔍
Topic on Extension talk:InputBox
I believe this would be covered by phab:T106948 ("Convert InputBox to use OOJs UI where possible"), but I'll ask some folks...
I believe that task would cover the issue of extensions/tools which desire to have consistent/pre-determined icons within buttons, see examples in OOjs_UI/Widgets/Buttons_and_Switches. So, after that work was done, additional work could be done on the InputBox extension to include a magnifying-glass icon on the type=fulltext/search/search2
form input buttons by default. However, changing the image manually for each usage of the InputBox, would run counter to the desire for standardization (for a more intuitive/learn-able interface). HTH.