Question: In our road we have a crab apple tree that has fruits but there have never been any small new trees rooting from the crab apples. Once, years ago, there were 3 crab apple trees within 100 yards but 2 of these were destroyed by some ignorant house owners. However in my 83 years of living within 50 yards of these three trees I have never seen any young new seedling trees. Some of the apples must have fallen onto the surrounding soil that is never disturbed. Will an apple tree grow if I deliberately plant some of the fruit from this one remaining tree?

Answer: 
Yes. You can learn a lot from this extensively reworked and thought-out answer to a similar question about roses: [How do I grow roses from seed?](https://gardening.stackexchange.com/questions/5548/how-do-i-grow-roses-from-seed)

Fruits of all kinds are attractive to wildlife, and are presumably picked up and eaten or stored in underground dens where the seeds never see the light of day again, or immediately eaten and deposited later in a remote bank. Seeds may well have built-in dormancy which ensures that they do not germinate for years to spread out the risk. But sometimes you can wait too long.

Just burying the fruit or seeds probably won't give you the hoped-for result. It's a matter of making it a special project and putting in lots of TLC to make sure the trees survive.
The short answer is yes. There's a lot of nuance to this though. If you select a fruit from the nearby tree that's ripe enough to have viable seeds (black or brown not white) AND they are harvested correctly from within the apple AND they go through the proper stratification process AND they get the right conditions to sprout then you're well on your way to *maybe* an apple tree.

I say *maybe* because the fruit you harvest the seeds from will not be the fruit your tree would grow up to be. Since nearly all apples require cross-pollination you'll end up with a hybrid fruit from the parent tree as well as the pollen from the pollinating tree. Could be something really awesome and you just created a new cultivar. Likely, however, it will be some sort of crabapple-type tree. Especially since you mentioned that you have crabapples in your area.