Deductible miles

Like everything in the tax code, it depends. As long as your home is not considered your principal place of business, you can follow this guidance from IRS Pub 463:

Home.

Home is the place where a taxpayer resides. Transportation expenses between home and the main or regular place of work are personal commuting expenses and are not deductible. A taxpayer can have more than one regular work location on different days. In your case, your "regular place of work" is likely the grocery store.

Temporary work location.

A temporary work location is a place where the taxpayer’s work assignment is realistically expected to last, and does in fact last, one year or less. If the taxpayer does not have a regular or main work location, transportation to a temporary work location is considered nondeductible commuting.

Second job. If a taxpayer regularly works at two or more places in one day, whether or not for the same employer, deduct transportation expenses of getting from one work place to another (one grocery store to another). If the taxpayer does not go directly from a first job to a second job, only deduct transportation expenses of going directly from a first job to a second job. Do not deduct transportation costs between home and a second job on a day off from the main job.

If you are delivering groceries to the customer, you would be going from the store right to their house. I'm not sure how your house would factor into this because you are not delivering groceries from your house; you are delivering them from the store.