My company is also all about giving back to the community. They support volunteerism in a few ways:
1. Employees who volunteer with a specific group can log their volunteer time in a company database. If you log even one hour in a quarter, you can apply for a $200 donation to your organization, selected by random drawing. There are some rules in place about what organizations qualify: must be a 501(c) 3 or a school; no political or faith based organizations, etc.
2. The company organizes and offers volunteering opportunities outside of work hours. They are voluntary, attendance is not monitored. They will also help your work team organize a volunteer day if you have a specific project in mind
3. The company also offers occasional volunteer activities during work hours - my team recently spent a few hours tying fleece throws for nursing home residents.
All of this is organized and managed by a dedicated team of community service coordinators, working closely with HR and legal to be sure everything stays legal and appropriate. They also make sure to publicize and thank the volunteers who participate, and do followup with the organization to see how the work benefited them, fine tune the experience, and develop ongoing relationships. *That's* how you get your company's name 'out there'.
Edited because the volunteer coordinators are not themselves volunteers; that is a regular full time paid position and has its own budget.