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: 
Not if you just plant a whole fruit, no, very unlikely. There is a way to do it, but you need to extract the seeds from the fruit in autumn by cutting it open, lifting the seeds out carefully, then removing any remaining pulp by flushing with water. After drying, they need cold stratification for up to 120 days - a description of how to do all this is shown here <http://homeguides.sfgate.com/propagating-crabapple-seeds-27860.html>, but you should bear in mind that Malus varieties do not come true from seed. The named varieties you buy are vegetatively propagated to ensure they are the right variety - from seed, you don't know what you'll get, other than it'll be a crabapple of some sort.
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.