We live in Tampa and go to Disney regularly with our 2 & 1/2 year old. We have a Bob and used to bring it when he was younger, but I always worried about someone taking it. Never had a problem, but I'm pretty familiar with Disney, and I don't think I've ever seen a place... at least not a convenient one... where you can lock a stroller. I would check with them before going if you really want to lock it. They are always moving the strollers to accommodate more, so I'm not sure that they would even let you lock it, and it definitely would be tough when it's crowded to try to find a spot. We now use an umbrella stroller, and it is so much easier, for navigation and for parking. I wouldn't leave anything of value in the stroller, either, just have a smaller bag with your valuables that you can take with you on any of the attractions.
I am also not sure if you will be able to drive to your family staying on property. Don't count on that, they check when you go through the gate, and you need a parking pass. It may work since you know someone there, I just wouldn't want you to be surprised if it doesn't. Parking is steep, $14 a day.
Disney has child care centers at each of the parks, and I highly recommend taking advantage of them. They have nice changing tables and places to feed your child. They have nursing rooms, as well, if you are still nursing your 6 month old. Microwaves to heat up your food, too, and anything you might forget... formula, diapers, etc, they have there (for a fee, of course). It's a great place for a break, especially since your kids are so young. They have play areas in the bigger ones, and a TV showing, of course, Disney cartoons. Look for them on your park map, they are worth going a little out of the way.
The best parks for your kids would probably be Magic Kingdom and Animal Kingdom. I think Epcot is a little too boring for the little ones, although the Flower and Garden Festival is going on right now, and our son really enjoyed seeing the characters sculpted out of plants & flowers. We did a character greeting at Epcot and it was a breeze to get through, very fast, esp compared to Magic Kingdom. You can also see the characters at Camp Minnie Mickey in Animal Kingdom pretty easily as long as you time it right (go when the Lion King show is in progress. It gets busy just before and after the show). At Magic Kingdom, there is an entire area just for the little kids, with things like Mickey's house, Donald's boat, a play area with slides, etc. It's been very busy the last few times we went, but hopefully it will be a little quieter for you since you're between seasons.
Have fun, it truly is great to experience Disney through the eyes of a child.