Two parts or two questions, not sure: Is it more efficient for a supermarket to send a van round to deliver stuff (eg a weekly/monthly shop for groceries) to multiple addresses, or for those people to drive (locally) to go and get it?
This depends on the area where you live.
If you live in an area where the houses are scattered around, the van has to drive essentially the same amount you would drive in your car. However, the difference is that the van driver is doing an 8-hour workday, and it may be the case there are two 8-hour shifts per day for the van, so the van must support 16 hours of continuous driving. At average speed of 50 km/h (realistic for areas where houses are scattered around), that's 800 kilometers of range requirement. An electric car can be driven around at energy use of 0.16 kWh / km, but a van might require as much as 0.30 kWh / km. So the van must have a battery of 240 kWh at least -- plus all batteries lose capacity over time so to make that van have an acceptable service life, you must account for 25% degradation in the battery at least so that's 320 kWh of battery requirement.
Do you know how much weight a 320 kWh battery has? A battery module (as opposed to individual cell) has perhaps 200 watt-hours per kilogram, so that's 1600 kilograms of battery. Not to mention the cost: at 150 dollars per kWh, it's 48 000 dollars for the battery alone (a van seeing heavy use may go through several batteries in its lifetime)!
In contrast, if you live in an area where houses are scattered around, your daily drive is probably less than 100 km, and 200 km will probably handle all driving needs in your area except the longest trips. At 0.16 kWh / kg, 200 km is 32 kWh, and factoring in 25% degradation, a 43 kWh battery (215 kg) is enough for you. The battery won't weight much more than an internal combustion engine, transmission, exhaust treatment, plus all accessories needed by those, plus the fuel tank, plus the contents of the fuel tank.
So, in a suburban area, an electric deliver van is simply not realistic. An electric car is (and you can charge it at home!)
In contrast, if you live in the middle of a city, the average speed of a delivery van is perhaps 20 km/h, and with regenerative braking and lower speeds the energy need lowers to perhaps 0.25 kWh / km. That requires 80 kWh battery, and taking into account 25% degradation, 107 kWh of initial capacity. This capacity is entirely realistic.
So, in the middle of a city, an electric delivery van is realistic. An electric car? Not so much, because if you park your car in the side of a street, there might not be a charging outlet nearby.
So, in the middle of a city, a typical car won't be electric and a typical delivery van will.
In the suburban area where houses are scattered around, a typical car will be electric and a typical delivery van won't be.
Also take into account the fact that in a suburban area the driver of a van can't optimize the route to combine trips (everyone lives far away from the city center and far away from each other), but in the middle of a city, the driver of a van will optimize the route to combine trips.