I'm not familiar with the delivery services mentioned above. Sounds like they'd be good for a large order though....
Where's the house located? If it's north of the hand ferry, I think the delivery option would be a great idea. South of the ferry you have a couple of decent grocery options. San Pedro Market is in the Del Rio area on the north end of town - off Middle Street. Island Supermarket is south of the airstrip a couple of blocks. It's also across the street from the Belikin and Coca Cola distributor. Buying the beverages from the distributor will save a lot compared to purchasing in the stores…. If they deliver to the area you're staying in, the delivery is made on a flat bed trailer pulled by a John Deere tractor. Where in the states can you see that?
Whether or not you use a provision service, you owe it to yourself to personally shop in a grocery store. It's a memorable experience.
You probably won't find all of the staples that you might want, but, you won't starve either. We've had mixed results in buying cereal on the island. Usually it's been fine, but, a couple of times the cereal was stale. There are a few places in town with a good selection of produce - I'm more familiar with the one on the ocean side of Middle Street near the south end of town. They're good and they also sell fresh squeezed OJ and they'll also shred fresh coconut for you. I've never been impressed with the selection or quality of meats in the stores - do a search on the board; maybe someone has better information than I do for meat. But, I've seen a number of posts regarding bringing in steaks, etc. for BBQ's.
Prices are going to be significantly higher in Belize than in the states. Products imported from Mexico, Denmark or New Zealand are generally less expensive than their US counterparts.
In my opinion, it's not worth schlepping a lot of groceries with you. Belize is very casual, so you can pack light. The adage of taking out half of what you initially put in your suitcase holds true. Why fill it up with groceries?
Take the teens with you shopping - they'll get a kick out of it. And, if the bottomless pits end up needing extra munchies, the night time vendors near the park are always cheap and good.
Have fun!