www.wysiwygwebbuilder.com

I am looking to add some simple forum to the site. No need for any fancy features (groups, categories, Emailing etc.) , just something simple forum. I know phpBB, bit it might be overkill, and I want something I can integrate with WWB site (possibly free or cheap).

Any recommendations ?

There is no forum extension as such. A forum generally runs separately from your site as a standalone.
Maybe the following from php jabbers may be of interest.
https://www.phpjabbers.com/php-forum-script/
or one of the following so cheap it is "free"
https://mybb.com/
https://www.simplemachines.org/
https://mylittleforum.net/index.php

BaconFries - thank you for your links. I knew one of them and I will check the other ones. One of my concerns is to maintain one user database.
I have never used WWB Login features so I am not sure it this would be possible to match WWB users table with forum table. I do some PHP development (mostly using minimal coding tools, simply because I am to busy doing other things), I would rather avoid writing a special code to update WWB tables with the forum tables, but if there is no choice I would do that.

Another concern is to somehow match the forum styling with the site. It does not have to be perfect, but at least on the color level...and I am not very experienced in CSS (that is why I sue WWB). If I get one of these to work I will post a demo here.

Thank you so much.
ART

Checked myBB and it looks pretty good. It has great documentation. The only challenge is to figure out how to sync the users table with WWB ?

I'm sorry, there is no option to sync users with a forum.
Most forums have a complicated database structure, so you generally do not want to mess with this via custom scripts.