Excellent idea tcm.
I was going to suggest an ignition coil driver circuit, they are all over google, just a 555, fet and a auto ignition coil, I've used a similar setup as the hf start on a vehicle based tig welder, vibrating relay contacts is a very similar system as used on the 1930's ford model T ignition, it worked for many thousands of them.
Strictly speaking for your application safety wise you really ought to have a burner control module, so that safety functions are included so you dont get the gas valve open without a flame, amongst other things.