When you talk about making grants to other “nonprofits,” I assume that you would intend to make grants to public charities, not other (c)(4) or non-charitable organizations.  If so, the answer is that you may, but the real question is why would you want to?

A (c)(4) can have purposes that include charitable activities.  If you can give the money away for charitable activities and consider that within your (c)(4) purpose, you could also make the grants directly. You would not have to pay a fee for the community foundation to manage the activity, you would not have to give up control over your money, you would not be limited in the types of organizations you could fund, and you would not be limited to giving the annual income instead of using some of the principal when appropriate.

I think community foundations are great things.  You might find the community foundation services sufficiently valuable so that opening a fund would give you the benefits you are seeking at a price you are willing to pay, particularly in vetting a wide variety of potential recipient organizations with which you are not directly familiar.  But you are already a tax-exempt entity and don’t need deductions for charitable contributions.  You would be giving up fees, control, and probably a significant amount of flexibility.  I would think carefully about whether I could fulfill my objectives by doing it alone.