We have mineral acres which are in our name but my wife would like to share the rental with other family members. Is there a way to use a 1099 form to pass the tax on to the recipient of the money so we do not have to pay the taxes and then deduct them before distribution of the money?

No, not unless they have ownership of the acreage.

If you "share" the rental income, you are giving a gift, NOT transferring tax liability.