Black labs are extremely social and very good family dogs. They are rambunctious by nature, but if yours is calm (at the moment!) that might be a good thing. They are pretty hardy so there's not usually a big problem with a toddler falls on top of them!
That said, you need to understand that you will now be housebreaking a dog in the dead of winter. Not the best timing. Figure out how you will take your dog outside at the first sign of peeing on the rug, since you will not have time to get your kids into coats and hats and mittens, and they are too little to leave inside alone! Puppies take a while to figure out how to "go" outside, so you need to spend a lot of time being patient while they sniff around, grab sticks, chase squirrels, and do everything EXCEPT what they are there for! And there's nothing more chaotic than having a 3 year old on the potty, a toddler with a full diaper, and a puppy with a full bladder! It's pouring rain, it's dark out, and the crazy dog needs to go out NOW.
A fenced yard is a good thing, but you can't leave them out there forever, and you still need to go out and pick up poop if anyone else is going to use the yard. (It's also a law in most towns, even if the poop is on your own property - it gets into the water table eventually and that's a bad thing.) The bitter cold weather is tough on puppies. Also, if you live in an area with woods, you have to worry about coyotes. I have a small terrier and the animal control officer told me to NEVER leave her outside no matter what. It's not as much of an issue when a lab is full grown, but still, the coyotes travel in packs and can work together to bring down a dog - our neighbors lost a small-ish lab this way. Heartbreaking.
I know how incredibly wonderful it is to look at a sweet puppy and know that you are rescuing one from the humane society. However, I have to say that you are better off making some sort of adjustment. First of all, someone else with older kids will scoop that puppy up quickly. Secondly, you will be better off if both your children are potty trained so you only have to train a dog! Or, you can rescue an older dog - even a 1 year old animal has had some housebreaking training, and they only need to learn which door to go to in your house. They have bigger bladders and can last a long longer! And they are past the chewing/teething stage. They're slightly less adoptable so you will be doing a greater kindness in choosing one. Finally, getting a puppy in the spring or summer is much much easier - it's more pleasant to be outside at 5 am or midnight (for you and the dog!), the days are longer, and the kids can be outside too.
Think also about dog hair and vacuuming! You will need to have a crate for the dog so it has a place to go to get away from the kids (it needs a break and it needs to nap) and so you can contain it as necessary. Puppies chew a lot so you can't leave it loose when you are not home or not able to watch it. Kids leave shoes and toys out, and the puppy sees them and says "Wow! That's all for ME?!?!!?" Little kids LOVE to play in the doggy's water dish too!
I would suggest you go to the local library and get a good video on dog training as well as a couple of books on the subject. Figure out how you're going to manage everything, and make a good choice.
Understand that I am an unrepentant dog lover - my terrier even sleeps with us (a huge "No-No" but she's 13 now and my husband wants her close to us! We love her terribly, but I'm telling you it is still a huge pain to go out in bad weather and watch her try to pee when the snow and ice are getting up between the pads of her feet and making her limp! Then I come inside and have to dry off 6 feet (my 2 and her 4) before she walks all over the house! So I want you to know what you're in for, and suggest that you either adjust the timing or the age of the dog.
Sadly, there will always be dogs that need rescuing - but that's also something that will work in your favor.
Good luck whatever you decide!