Yeah, it's a weird catch-22. But I think there are a lot of smaller businesses (maybe mom & pop shops) that would be totally fine saving on cost to employ a student (and depending on the student getting some pretty good results). Tons of students are already the "computer kid" (or something similar) that all tech support/website help gets funnelled to...
Maybe give students a way to submit leads? Like if a student has a family member that wants to make a site, but they don't want to do it themselves, you can give them something (not sure what) for submitting it to the site?
Yeah, basically -- you should be able to run a ton of it without building it yourself (for speed, and of course tons of open source tools already exist), but the idea is basically for different types of students to find each other.
When I was thinking about it, I roughly broke it into some of the types of roles you have in a startup
Engineering/Programming
Design
Business
X-Factor (for those that don't fit in the narrow categories I've suggested)
Then students can sign up, add their resumes, and post ideas that they want to collaborate/get help on, and see if they can attract interest from other types of people.
The site would then provide collaboration tools (gitlab for code, general chat/team management with phabricator for example, etc) to the group.