As you must know, the collector current of NPN transistors is roughly proportional to the base current. You need to increase the base current. Different NPN transistors have different gains (basically hFE), but be aware the gain varies widely depending on each individual transistor and operating conditions...
Here one way to increase the base current would be to lower the value of the 22k resistors (but you'll need to adjust the capacitors accordingly).
You could also use darlington NPN transistors instead. But this basic astable circuit with NPN transistors as is is not ideal for "high" currents.
One approach you could use here without redoing the whole circuit with MOSFETs would just be to add ONE MOSFET and drive its gate with one of the outputs of the multivibrator (collector of either NPN transistors, put a series resistor back from collector to 5V.