The first question should really be: is this lathe capable of turning bowls? What is the swing capacity (the distance from the center of the spindle to the lathe bed)? If I remember correctly, Buffalo tools are cheaply made and light weight, not really made to spin larger pieces of wood (6"-12" diameter). Mounting a piece of wood that's heavy or out of balance on a light weight machine can be very dangerous, be careful!
That being said, you don't really need a chuck to turn bowls, you can start between centers and mount with a face plate and glue block. Using a glue block is more stable and will have less vibration. Gripping a piece of wood with a tenon is like gripping a sponge, the wood compresses.
My suggestion would be to either join a woodturning club and get some instruction from one of the members or taking a turning class (WoodCraft offers classes) before attempting to turn a bowl on your own.