I have the chip & spare key, could this work to start my car?

Hi there. I have the original chip for the primary key of my car. The key
somehow separated from the original fob and fell into the snow, so I have the
fob/chip but not the actual key to put in the ignition. I also have the spare key,
but it's my understanding that the spare key isn't programmed so I can't start
the ignition with it. It only unlocks the car.

My situation is unique in that I have the original chip still, and the spare key.
Since I have both, could a lock smith potentially sync the chip with the or a
new key? Or could I do this myself? Thank you.