We've long known thát Lego people don't reproduce in the same way non-Lego humans do (see rumguff's excellent answer), for all the ages that have passed in Legoland, no one ever knew why. Now that another age has passed, even since this query was asked, new revelations out of Legoland confirm the answer that modern Lego people -- Lego minifigurae manihabiles -- do in fact reproduce, just not in the way we might expect.
First some history: We know from the most ancient of records to be found in Legoland (discovered recently in the Bookshop, set 10270), that there were, in fact, three races of Lego people.
In those days there were giants in the land... Back in the earliest epochs of Legoland history, all Lego people were big. Lego magnitestae, the so-called "big heads".
And as we can see, Lego people at that time did in fact reproduce and grow old. During this archaic epoch, certain environmental factors spelled the downfall of the "big heads". First, and although we have little material evidence from this time in history, we do have tantalising hints of multi-limbed and tentacle-limbed monstrosities being born. The relative sparsity of basic building materials in the archaic age also spelled doom for these giants, as even a family of five simply could not procure enough bricks or base plates or roofing shingles to create even a modest cottage, to say nothing of banding together in communities.
And so the archaic age with its reclusive giants came to an end, outcompeted by the most unlikely of evolutionary freaks.
During the short lived middle epoch of Legoland history, a new kind of Lego person was created from the potential brick, Lego minifigurae rigidae. These bizarre people were much shorter than the "big heads", and were, in many ways, more frightening than the great monsters of old.
With their expressionless faces and immobile trunks, yet somehow able to braid their hair and wear their caps, they dominated Legoland for a short time. Yet somehow they were able to rebuild Lego civilisation, undoubtedly drawing on the ruins of the "big heads'" earlier attempts.
Their time of dominance was short lived, however, as the middle epoch evolved into the younger epoch and the rise to dominance of the minifigure proper was accomplished.
From the medieval period...
to the space age...
Lego people kept advancing, but one thing neither the rigid nor the articulated kinds of people could do was reproduce -- unlike the big heads, we see no children in these later epochs of Legoland history!
Not even the short-lived evolutionary throwbacks that arose at this time, Lego magnatesta basica, had children.
And at the time this query was written, that was the state of things.
But since that time, as was mentioned earlier, new revelations have come out of Legoland. There are now known to be Lego children and Lego babies!
It's unclear how they are formed, as their brick matrices vary from the normal, though it is known that, as with adult Lego people, children and babies are brought fourth from that mysterious place called the Brick Box, having been formed from the primordial brick. Some believe that Lego pediatricians have something to do with the process, but there is no clear evidence for this.
At least this much is clear: as it was towards the beginning, it has always been the Plan for Lego people to live in family relationship and exhibit varying ages among individuals of society. It is simply not until the modern epoch that Bill and Mary's descendants have been given this gift by their Creator.