In a right triangle with integer length sides, the hypotenuse has length 39 units. How many units is the length of the shorter leg?

39^2 - 36^2 = 15^2

So, the shorter side = 15 units.

Notice that  one possible  "Pythagorean Triple" right triangle is  one with the sides  5 - 12 - 13

Any any  multiple  of these will also be a Pythagorean Triple...so....this implies that

5(3)  -  12(3)  - 13(3)   =

15  -  36   -  39   will also be a Pythagorean Triple

So....the shorter leg  =  15 units