Talk

We had this issue but found this place which was fantastic for all of us, it isn't near Port Elizabeth in South East of South Africa www.kariega.co.za/experience/kids-on-safari/

What time of year are you looking at as this will impact on which countries to look at.
This camp in South Luangwa is great, kids under four years stay free think you just pay about £30 a day for food and game drive for them.
They have family chalets and an amazing tree house.
www.flatdogscamp.com/
then you could go to Lake Malawi kids could try out horse riding, watersports, playing with the local kids on the beach. You could stay on one of the islands
www.safari.co.za/Likoma_Island-travel/malawi-safari-kaya-mawa-lodge.html

Or self drive in Kruger South Africa if on a budget. Rent car,Cheap stay in park accommodation, buy a family wildcard for entry fees. We do this every year.
www.sanparks.org/

We went to South Africa in April and stayed at Hlosi Lodge with our 3 and 1 year old - I would highly recommend it. The staff were lovely, and the ranger even allowed the kids to come on a couple of short drives with us, which we weren't expecting. We all loved it.

Sorry - link here
www.hlosilodge.com