I think the answer will depend primarily on the organizational name the donor puts on the check.  If the gift is given solely to your organization, even though the donor recognizes that parts of it will be used by other charities for the service, I think it will be legally sufficient if the donation receipt comes from your organization alone.  This is particularly true if the ultimate allocation of the funds is not predetermined and no individual donation is earmarked to any other participant in the group.  (I am assuming that your “partnership” is not a legal partnership which would have its own exempt status and is merely a shorthand expression for a type of collaboration among the providers.)

It wouldn’t hurt to give information in your acknowledgment about the service you intend to provide and the other organizations that will participate in the work. You could thank donors on behalf of your organization and all of the other members of the group.  Donors may be quite pleased to get one acknowledgment letter instead of several.

However you acknowledge the gifts, you should be sure that all of the groups are pre-registered to solicit contributions if you are soliciting in any of the 38 states and the District of Columbia that require such registration and they would be required to register if soliciting there individually.  This fundraiser will be at least an indirect solicitation on their behalf.