Would a failing igniter cause my furnace to blow cold air?
Could be. Could also be the flame sensor not working (if the control system can't tell that the furnace is lit, it will assume it's not and turn the fuel off.) Could be a nozzle clog (since you didn't specify your fuel - that's more of an oil than a gas problem.) Could be an issue with a power vent, or a power vent sensor (again, since no details provided, may or may not apply to your furnace.)

Unless you really know your way around them (and quite often even then) it's often a good idea to have a pro in to deal with fuel heating appliances - they have knowledge of the systems, and esoteric tools that don't make sense for a homeowner to purchase, such as the exhaust gas analyzer that allows adjusting the fuel/air mixture for best efficiency.