Closest costco was on Harbor Blvd. in Fullerton - not too bad if you have a car, no way if you have to walk. My son and I took the bus it took over 45min each way - but we met some interesting people
It was very memorable for my son - talks about risking his life on the bus in Anaheim - can you say spoiled! I still laugh when he talks about it!
There is a Target for non grocery stuff about 5 min away from the park on Harbor Blvd and then just a bit farther down (maybe another 3 or 4 minutes) is a Vons grocery store (simialr to Safeway, if you have one of those- even my NW Safeway CLub card worked here!).
Google map grocery store near Harbor Blvd in anaheim and you'll get results
The Target at Lincoln and Euclid is expanding their grocery section, with a target opening date of March 11th. They had a pretty good grocery section before, but I understand they're adding produce and a meat section.
Much closer to DLR is Food 4 Less on Katella about a mile - no frills, bag it yourself, but good prices.