Doug is right, the only way to remove the tappets is through the camshaft bore.
The Cummins kit has a tray that is like a tube with a filet missing from the top half and a cap on the back. There are also dowel rods that are a little longer than the length of your pushrods. They are slitted in one end and usually have a sanded taper. I don't recall the exact size, but I want to say they are about 3/8"
The kit also has one long dowel rod with a cuphook in the end. The final piece is a braided sting. (Think surveyors nylon line- not cotton or polyester) and a rubber plug to fit into the end of the tappet.
The dowels must be used to hold all the tappets up in thier bores, so they must be friction fit into the top of the tappet. Whenthey are held up, you place rubber bands to hold the dowels together. The pressure holds the rod and tappet from falling.
If after the cam was out and the tappets were to be removed, they would be droped into the tray in the kit one at a time. I used a little magnet pen to fish them out, but the hook works as well.
The hardest part of the install is reinstalling those tappets. It takes a while and has to be done right. If you drop one, you are fishing or pulling the pan.
My stock tappets looked like new and I questioned why I bothered swapping them at all.
Dave