Contributor user can't post HTML

Have they tried entering their content under the HTML tab? If they are writing HTML code in another app they copy and paste it into the Post or Page under the HTML tab all of their markup will be there.

Roles and Capabilities

See Capability vs. Role Table > unfiltered_html

You may be able to easily define custom capabilities to specific rolls or specific users by using a plugin.

Here is a quick example of what you might want to look for. User Role Editor

You’re welcome.