I think this is a great idea and is exactly how capstone projects were generated for my entrepreneurship major.
A form was available for local businesses to request help for strategic planning. The owner would submit their proposed project and expected outcome. The professor gave students a survey to get a general idea of sectors and topics the students were interested in. He would take the survey results into account and select enough companies for every team to have 3-5 people.
A brief of each company was written with the general project scope and distributed to students. Students would rank the top 3 projects they wished to work on and were then assigned teams.
Mentors with experience in each businesses particular industry would be assigned to teams to provide expert opinion.
I think the problem would be the need to have projects that were "capstone level" if you intended to have computer science majors doing this. Apps, scheduling widgets and other technical projects may satisfy this but website creation or redesign would not.
A better avenue might be to work with coding bootcamps; as they would have mentors in place and are focused on more entry level websites and applications.