car will run if o2 is bad and u will get a check engine light. should not stop it from cranking.
i would replace the plugs,wires and distributor cap with rotor button for starters.

the thermostat should not keep it from cranking either and if you are not sure just take it out and try it without the thermostat. just takes your car a while longer to heat up that way.

i would say either spark sys, or ignition sys ie... switch or relay.
cheaper to replace spark sys though and never hurts.