If I take a cab from LGB:
Are there cabs that hold 6 people, a van?
Will they stop at a grocery store - is there one along the way and does it cost more if they stop?
There are both mini-van and larger van taxis, but whether one will be at the airport when you get there isn't guaranteed. Only once have I traveled with a group over 4 people (4 adults, one 3 year old) and the mini-van taxi worked great for us, (including the car seat, stroller, etc). We didn't schedule a van taxi at the airport because they're almost always there (I think you find more van taxi's at the airport than around the city because the chance of needing a bigger vehicle is higher there) The last trip we needed a van and there were 5 cabs, two of which were mini-van taxi's and we just told the guy queing the taxis we wanted a van and he motioned for the mini-vans in the back of the line to come up. I imagine most mini-vans allow you to sit in the passenger seat also so you could fit 6, the only question is if you can fit your luggage in the back of the min-van or on your laps.

One one trip back to the airport we ended up in a 7 seat full-size van.
I'm sure they will stop if you want, but you can be pretty certain that the set rate from the airport won't be honored or it starts as a "new trip" from the store. You could always take the $45 taxi to the store, have someone wait outside with your luggage while you shop and get a new taxi to take you to the resort, but that would be a bit of a hassle. For some reason I have the idea that there are delivery services that will pick up groceries or other items and deliver them to your hotel but I'm not sure what the cost is.
Also, there are two routes from LGB to the DL area, the city streets way (considered the short way) goes by a few stores if I remember right, but I never pay that close attention so I couldn't say what they are) and the freeway way (the long route). The city streets are almost always faster and it's a cheaper fair FYI.