Yes, there are some genetic conditions where puberty is delayed and has to be brought on by certain drugs. Puberty is driven by hormones so if something damages the glands that produce them, it can be delayed. Generally though these conditions are incredibly rare, and because everyone is different, there is huge variation on when people hit puberty anyway.
I actually study one of these conditions, called Kallmann Sydrome. There are a couple of hundred nerves in your brain that produces the hormone that kickstarts puberty. If these nerves don’t develop properly then puberty will not start.
There are various genes that are needed for these nerves to grow. We have a mouse that lacks one of these genes and we are studying it to understand how it works, how it helps the nerves to grow, and ask what affect losing the gene has on the body of the mouse and on its behaviour.