The answer to the first part of your question probably depends on how the donations are being solicited and contributed.  Does the church solicit or receive the gifts for a food pantry or for your food pantry?  Your question seems to suggest the latter.  If the gifts are “earmarked” for your organization, they are deemed to be given directly to you.  In that case, you should give the gift receipts (noting that the money came through the church) and you need the names and addresses of the donors and the amounts donated in order to be able to do so.

If the gifts are given to the church and the church controls what food pantry gets to use them, the gifts are for the church and the church should acknowledge them.

The issue most frequently arises when the ultimate recipient of a gift for charitable purposes is not an exempt organization and the donor wants to make a gift to a recognized charity (and not to a non-exempt entity) in order to claim a charitable contribution deduction.  Since you are a recognized charity, that isn’t necessary here.  The gifts are deductible in either case and don’t affect the public support rules that would require either of you to be classified as a private foundation.

Even if the church is claiming the gifts and properly acknowledging them, I doubt that they are soliciting for “anonymous” donations and there is probably nothing that says they can’t give you the names and amounts so that you can thank the donors directly.  You should try to work it out with the church, particularly if the gifts are designated for you.  Donors normally don’t get upset with an extra thank you.