Last year I took a class that went over resume, cover letter writing, and interviewing skills. One of the topics of discussion was about the importance of tailoring your resume to the job you are applying for. You should never just have one resume and use that for every job you apply for. The resume should be tailored to the job description for the position you're applying for.
We were also taught only to include the past ten years worth of skills related/work related history. Unless it was something spectacular or really special like peace corps service or teaching language in another country or volunteering abroad helping to build homes or something like that.
If the delivery job is related to the job you're applying for, then include it but explain honestly why it didn't work out. If it doesn't apply and you are applying for a different line of work, leave it out. If you get to the interview/background part of applying for the job, you can explain that you left it off because it didn't relate to the position you are seeking AND it was over ten years ago.
I was "just a mom" for 20 years and recently made the transition from focusing on my kids (who are now grown) to now focusing on a career. One of the best things you can do is volunteer, actually. And I don't mean at the kids school!
Find something you're passionate about and volunteer there. It's one of the best ways to get your foot in the door, and then gain work experience and references that begin to build you resume. When a paid position opens up (and they will at nonprofits as turnover is typically quite high) apply for it, even if you don't necessarily have all the "hard skills" that are listed in the job description. If you show up on-time, clean and sober, are reliable and hard-working, and people like you and you get along with everybody - employers are willing to teach you the hard skills if you've proven to them that your soft skills are strong.
Good luck! I know it's not easy and that transition can be a little overwhelming and emotional. Be easy on yourself and celebrate even the small stuff.