Yes. Yes you can. But as it’s a commercial font then you can’t host that font yourself unless you have expressed consent from its creators. This is to prevent other users who haven’t purchased the font file from taking it from your web server and using it themselves.
The easiest way to use it is just to sign up for a fonts.com subscription and then you can use the webfont version of Avenir on your site. You can sign up for a free account which allows you to use it with some restrictions, or a standard account is just $10 a month – which also gives you access to their full library of fonts. Not a bad deal.
I always have a tough time finding out information about licenses for different fonts, but it’s important to do so (especially with a big player like Avenir) so that you don’t encounter any legal ramifications down the road and get sued.
Here’s some more information about using Avenir in particular: http://miksovsky.blogs.com/flowstate/2010/08/lessons-switching-our-web-ui-from-stock-fonts-to-a-proprietary-font.html