Our corporate wiki is Mediawiki. I have no problem to put iframe into my site to refer for some article on wiki.
But my own site have a lot of widgets and own style. I don't want to include Mediawiki navigation/search/login widgets, logo image.
Is it possible and how to get Mediawiki page contents without widgets (only article body)?
Yes, it is. You'll probably want to use the action=render url parameter, for example: http://en.wikipedia.org/w/index.php?action=render&title=Main_Page. Note that the stylesheets from the wiki aren't included, so you'll need to copy the relevant rules to your site's css files. See also this.