Another option would be to get a Garden Grocer delivery to your hotel but you would probably need accommodation with a fridge etc to make that work. Onsite is my preference as the transportation is regular. I have heard (not experienced) offsite shuttles can be a bit hit and miss. With your children, I would want to be able to get back to my accommodation without long waits and hassle.
Was about to post the same about Garden Grocer!! You can order things like bottled water (you would get 24 bottles for the same price as 2 or 3 if bought in the parks), cereals, snacks, other "heavy/bulky" items, and then use the resort shops to buy milk for cereal.
There is also a quite well stocked HESS garage across from Downtown Disney that is much cheaper than the resort shops.
Regarding staying off-site, most of the hotels that offer a shuttle service do so on a limited basis i.e. possibly 2 or 3 return journeys to each park, or maybe only to one park and you would then have to use Disney Transport to get to the other parks. If you stay in the Lake Buena Vista area (along Palm Parkway), Lynx Bus #50 operates every 30 minutes to bot Downtown Disney and the Transport and Ticketing Center (across the lake from Magic Kingdom) from where you can get transport to all the parks. The last #50 bus leaves Disney at midnight.
If planning a few days away from Disney, it is possible to rent cars for a couple of days and pick them up from the Disney area, or at hotels in the Downtown Disney area. However, as noted in a few posts above, if you use a car to visit Universal or the other parks, it will also cost around $15/day for parking.