We have a few different options for this in our area (online or phone ordering and delivery) and the delivery fees range between $2 and $8.95. Many many groceries have begun offering this as a service..so perhaps ask around if you live in a metropolitan area. (Heck, we live rural and we’ve been able to get grocery delivery for $2.)
Wal-mart, Peapod and many other retailers provide this (at least here in the US).
If your family member is elderly they may also qualify for a free (volunteer) service to do their shopping if they live in an area that provides it.
We have a service like that for seniors in my area.