I had an interview at a university for a tenure track position. They decided not to give me an offer, but I met a professor there (for the first time) that supported me as a candidate. This professor saw my application, went to my talk, and was not on the search committee, but participated in my campus visit.
I corresponded with this person through e-mail, and the phone, about my career (not just about the position).
Since I didn't get an offer for a job I wanted this year, can I ask this person if they would write me a reference letter for my job search next year?
You can ask the person and there would be no conflict of interest in doing so. While this person might have a good understanding of you research, teaching, and background from the previous job process, they generally will not know you well enough to write a strong letter of recommendation. The best letter writers are people who have known you for years and have worked with you on a daily basis.