You need a grid six cells high and seven wide. Each cell needs to have at least 3 states. (Empty, Contains A's piece, Contains B's piece.) You can accomplish this with lights, like so:

Note that you will need seven columns, so you will have to set up seven distinct copies of columns A and B. You only need one copy of column C, but you will still have to wire it and the reset button to each of those seven pairs of columns.

Column C is responsible for keeping track of whose turn it is for the logic.

The first chunk of the main columns is responsible for determing that:

1. You pressed the button for this column
   
2. It is your turn
   
3. There is still space in this column.

If those are all true, the Not Switch and And Switch immediately below will be triggered. Column C will also be notified to toggle whose turn it is. The Not Switch and And Switch are intended to shorten the signal to exactly one frame. If it wasn't shortened, multiple cells would light up, instead of just one.

Each chunk thereafter is responsible for one cell. The Not Switch and the And Switches are responsible for determining that:

1. This cell is empty
   
2. The cell below has a piece in it (if applicable).

If those are both true and the column was selected, both nodes of the Latch Switch are triggered. This sets the Latch Switch to On until the reset button is pushed. The Latch Switch tells the appropriate pieces that the cell is occupied, and turns on the associated light.

The Not Switch in the bottom chunk also identifies when the column is full, and notifies the first chunk to prevent wasted turns.