The origin of the regional pronunciation of the word "ask" as "axe" has less to do with color or "ebonics" than it does with the area in which you were raised, or from which your parents came. The propagation of this has to do with subcultural trends in manners of speaking (which is sometimes identified as ebonics, but this goes deeper than that).
It was originally a Southeastern and Delta regional pronunciation of the word that you are hearing. If you wonder why this has travelled outside of the original regions with black people is that, due to needs of the manufacturing regions for workers to replace the whites serving in the then segregated military, there was a tremendous movement out of the south to places like Detroit, Cleveland, Chicago, Philadelphia and New York during the wars, and again during the 50's and 60's when there was another wave of ethnic migration to western cities by people looking for new opportunities away from Jim Crow in addition to the same reasons that everyone else seemed to want to go west including large numbers into Los Angeles.
The dialect of the original transplants carried over in some ways with their children who grew up in black neighborhoods that began to develop as folks were either redlined into their areas or moved into them due to economic or comfort pressures. These cultural subgroups kept their language inside of their own communities and had this reinforced by some who used the different pronunciations to make clearer the differences between them and others.
Now if you ever travel to the regions you may hear a white Cajun or a south Georgia farmer say "Lemme acks 'is feller a question..." and you'll have a better understanding why they are speaking this way.
Also, the next time you want to understand something about the topic of dialects and regional or ethnic pronunciations look into doing some research with some professionals who can give you better insights than we can without taking the hits for asking a question and paving the way for the reactionary flame wars.
Heres some sites for you.
UPenn source with excellent offline bibliography
There are plenty of other sources as well.