Are the courses for each corporation in different categories (can they be) or are the users actually using the same courses but you want them to see different logos?
In the first instance either David or Emma's solutions will work - a different theme for each category, or a little bit of css that targets the categoryid in the body class, such as (Sorry, I don't have squared theme installed at the moment and I'm not on a computer with access to my moodle, so the example below is illustrative only and not the actual code for using in squared - I'll try and find that later if you cant find it using something like firebug, but the use of the categoryid class will allow you to target any css selector below, or including, body with customisations for that particular corporation, so the exact css isn't necessarily relevant to the example)
.header logo {background: your_background_image_styles_css_for_default_look;}
.category-2 .header .logo {background: your_background_image_styles_css_for_corp1;}
.category-7 .header .logo {background: your_background_image_styles_css_for_corp2;}
If the latter, you may need to do some additional coding to make use of a user profile field that identifies the corporation the user works for and allows that to add a body class, then target the css using that rather than the category. You may also need to use this method if you want them to see their own corporation styling/logo on pages such as their own profile page which are not in any course or category.
You would add something like this in the early part of the relevant layout files
global $USER;
$bodyclasses = array();
if ($USER->institution == 'corp1' ) {$bodyclasses[] = 'corporationone';}
if ($USER->institution == 'corp2' ) {$bodyclasses[] = 'corporationtwo';}
Then the css would be something like
.corporationone .header .logo {background: your_background_image_styles_css_for_corp1;}
.corporationtwo .header .logo {background: your_background_image_styles_css_for_corp2;}
Hope that gives some help
Richard