Can i be removed from driving job, when i use my own car and own insurance for previous dwi conviction? 5 years or more ago?

Asked in Kingston, NY - about 3 years

i work for a big box home improvement retailer, have been with company for 6+ yrs. had a dwi over 5 years ago. back in december 09 they announced a new position doing in home sales. i was given the new job in january. february started training and the job began end of february, last week the corporate office decided to do mvr checks. now because i have the dwi showing up i am told that i cannot have my job. in this job i use my own car and my own insurance. i get paid mileage to reimburse me, can they do this after i have been given the job and have been doing the job? and i am not using their equipment or insurance. i just bought a newer car for this job and have even spent $700+ on a new wardrob for this job, my last position has already been filled by someone else.