I work for a private non profit that is located in a downtown district. The parking that is provided for employees is the public city parking lot. If an employee falls in this parking lot and gets hurt when coming/going to work would it be a workman's comp claim, is the organization responsible?

No, it is not workers comp, since they are not at work yet, They are still on public area. If they were being sent somewhere while working, then it would be a workers.
They are not covered till the start of their work day.

If the city was at fault with some unsafe condition, you may try to sue them, but most likely it is just an accident or the person not watching where they were walking ( their fault) and there would be no claim at all.

Not a WC claim as you were not technically at work yet. You were going to work. Now had you had that accident while you were at work and had to run an errand for the company that would count as a WC accident as it happened on "company time".