How can I sequentially power 3 coils?
Some 74LS logic would do this.  Specifically a 74LS590 counter and 74LS138 demux, paired with a 555 timer and a couple transistors.

Simply wire the 5th output on the 74LS138 back to the Reset on the 590 to initiate a reset after the first 4 pins have been sequentially energized.  Or use the 5th pin as an enable/disable bit, to stop the timer after one run.

Depending on current requirements, you could use 2n2222 transistors and flyback diodes, or something more heavy-duty like Fairchild's FDD8447L MOSFETs.  

Add the requisite passives and a 5v regulator for the 74 series chips, and you're good to go.