BuddyPress.org

@thebigbonanza – BuddyPress is overkill for what you want to do. This is what I’d suggest you do with your $700 per your notes above

1. Find a web development tool (free or nearly free) that will allow me to create and customize a social website that a dummy (me) can manage.
– Get a free account at WordPress.com, pay for upgrades to
a) get your own domain name pointed to your free WordPress.com account
b) get a premium theme installed (if you don’t like the offerings at WordPress.com)

2. On this site, I’d like for visitors to be able to upload their own videos of them eating my pizza, talking about it, etc, comment on posts, vote for fave videos, and forward posts to their friends via email, FB or Twitter.
– You’re going to spend a lot more than $700 if you’re going to allow people to upload videos in your own server if you choose that route, and you will need to upgrade your plan again at WordPress.com to allow others to upload videos there at limited upload space at that.
I suggest that you ask your friends to upload videos at Youtube, Vimeo, etc and to send you the video URL. Post that URL to your wordpress.com and enable free Ratings and Poll systems, then add free widgets for Twitter et.al on your sidebar.

3. I’d also like the site to prompt people to sign up for email alerts throughout the contest
– add free Subscription widget to sidebar as well so others can follow your blog.

4. link to PJ’s online locator of PJ stores and online ordering system.
– Add link to PJ’s online locator in free Text Widget you can add to sidebar

There are free tutorials to manage that WordPress.com blog at http://wordpress.tv and there is free assistance at http://en.forums.wordpress.com/

Then after the Papa John’s contest in August, if you want to go full social networking with own server and move over all your posts in info from your free wordpress.com account, then go BuddyPress.

Good luck.