The circuit is running slowly so the fets could be fed directly from the 4047.
The timing components for the 4047 should be sized for 120Hz or so. The frequency must not be so low as to put the transformer into saturation.
F=1/(4.4*R*C)
Normal fets need more than 6V to turn on fully, 10 or 12V is good. Feeding the 4047 from the drains via diodes gives about twice the input voltage. It is only necessary that the fets should be able to pass some current at 6V input to start.
You will need a bridge rectifier on the output to replace the second set of contacts, this can be made from individual diodes (1N7007) or from a moulded bridge (SKB1.5?).
You should check that the fets do not get hot and you may need some extra capacitors on input and output if you get interference.
The original vibrator will have a dead time as the contacts change over. There is often a capacitor to tune the primary so that the voltage reverses ready for the next contact to be made. This capacitor will not be needed.
A fuse is always a good thing. Polarity is critical, you may want to fit a fat diode to ground after the fuse so that the fuse blows if the polarity is connected wrongly.