Will working out during puberty keep you from growing any taller?!

Will working out during puberty keep you from growing any taller?

rumors are that if you workout around early teenage years you will saty short is that true??

No. I assume this is healthy working out, not overstressing your body and staying in-touch with how you feel secondary to what you did.
The reason one becomes taller or not so tall is secondary to the growth areas of your "long bones". (These are not for instance, "flat bones" found- say in your skull.)
In long bones, including toes, metacarpals, femurs, etc. there is an area called the 'epiphyseal plate' made mostly of cartilage. (When you were born, all your bones were mostly cartilage with a little area called the ossification center.) This epiphyseal plate serves as the place were intense cellular activity is occurring. In the case of growth, it is the deposition of bone cells (there are three types) into this area and the resulting calcification of the plate. The amount of bone cells are determined by genetic coding, lifestyle, and activity (to a much lower extent). It cannot be destructive activity such as trauma and the like. Once calcified, bone does not grow longer, it may grow possibly wider, but not longer. Assuming no injuries are caused by "working out" the determining factor for continued bone growth is multi-factorial. The important ones for adolescent growth period are: genetic predisposition and lifestyle choices such as: nutritional status which means including normal amounts of calories, vitamins, minerals and a healthy activity. These are the most important factors, there are more but are far beyond the scope of your question and my answer.
So, assuming you have the genetic potential to grow tall, including both at the torso and lower extremity areas, one needs to make sure that the genetic information is maximized by providing the cells that deposit into the epi plates are maximized also. Activity is important so as to address the question of damage to the body by unhealthy behavior such as avoiding toxins that are common to all of us, e.g. avoidance of tobacco, including passive smoke, (decreases oxygen flow to bone cells-DNA changes), alcohol consumption (possible changes to DNA structure), hormones that are not produced by you, say potentially if female, the use of birth control pills (but there is little scientific evidence for that) and other well known substances known to be detrimental to health in general.
You can have the genes to grow seven feet tall, but if you do not maximize your correct intake of calories, nutrients, vitamins, oxygen flow, you will not make it to seven feet. On the other hand, if you have the genetic info to be four feet tall, nothing you do can change that and get taller. Hanging from the ceiling by your feet all day will not help a bit. That is your maximum height -- period. (Growth hormones are an exception, but that level is DNA encoded. Doc's can change that if needed but that is an exception to my answer.)
I hope this is a bit helpful. To grow to your maximum, live wise, eat well, avoid excessive work outs (damaging via micro-trauma), and have joy.