If you are staying at a hotel you can grab a pool/beach towel from the hotel.
Do not recall seeing umbrellas at ABCs and Longs, but all the others listed above carry them.
There are a couple of places to buy a plate lunch in Wainamalo, the chain L&L BBQ and Keneke's. We usually carry out from L&L but the best choices are hot meals and may not work for a picnic.
There is a small super market, Shima's in Waimanalo with a good poke section, they also make poke bowls. You could get a few kinds of poke and other picnic items, but they would need to be kept cold until lunchtime. Shima's has the usual supermarket stuff if poke isn't your thing. You will feel like a local shopping there.
Whatever you decide, it is best to have a way to keep cold foods cold until lunchtime. You can get a small styrofoam cooler and a bag of ice and keep food and drinks cold. If we want to just keep small food items cold, we freeze a couple of water bottles in our hotel mini fridge and get an insulated carry bag from the ABC store. The bottles and the bag make a quick small cooler.
There is also a good ice cream place in Waimanlo, Dave's. It is across the street from Shima's in a small strip mall and has locally made ice cream in tropical flaors, my favorite is coconut macadamia. It's good for a post beach stop. There is a shop in the strip mall that has locally made items and is also worth a browse. It's a good place for souvenirs that are not mass produced and cookie cutter. Unfortunately I can't remember the name but is easy to find.
I believe it's East Honolulu Clothing Company...We missed it last year, because it sounded so tacky, then I looked at their website, and I was kicking myself....