There is actually a bug in the Header web template.
The class option as per @Dean_Warren should work, but the logic is missing a liquid else just after that class bit
this section of liquid in the Header web template is what makes Dean's class option work
{% if link.image %}
{% if link.image.url %}
{% if link.image.url.first == '.' %}
<span class='{{ link.image.url | split:'.' | join }}' aria-hidden='true'></span>
{% endif %}
but there is a logic bug which means it won't render the image to the width and heights specified if it is a web file image as there is not an else to the .first == '.'
It should be something like
{% if link.image.url.first == '.' %}
<span class='{{ link.image.url | split:'.' | join }}' aria-hidden='true'></span>
{% else %}
<img
src='{{ link.image.url | escape }}'
alt='{{ link.image.alternate_text | default:link.tooltip | escape }}'
{% if link.image.width %}
width='{{ link.image.width | escape }}'
{% endif %}
{% if link.image.height %}
height='{{ link.image.height | escape }}'
{% endif %}
{% endif %}
If you add the missing else, it will render a web file image eg. can use "/logo.png"
(the IMG tag I have used is what is several lines down in the Header template)
Edit: suspect some Dev edited the Web Template and added the liquid endif after the span class, when it should be further down (the existing else makes no sense - as it is basically saying if there is no url then create an img using the nonexistant url)
