...but evidence points to the Supers being born with their powers.
I'll start off by saying it is quite evident from the first film that the "second set" of Supers were born with their powers, meaning Violet, Dashiell and Jack-Jack. E also seems to hold to the same assertion that two Supers having children should produce a Super.
Edna: Shh! Darling! Shh! I cut it a little roomy for the free
movement. The fabric is comfortable for sensitive skin, and can also withstand
a temperature of over 1000 degrees. Completely bulletproof. And machine
washable, darling. That's a new feature.
Helen: What on earth do you think the baby will be doing?!
Edna: Well, I'm sure I don't know, darling. Luck favors the
prepared. I didn't know the baby's powers, so I covered the basics.
Helen: Jack-Jack doesn't have any powers.
Edna: No? Well, he'll look fabulous anyway.
With this being the case it would mean that the powers are genetic which would imply that they are born with their powers. This assertion seems to hold through for the following description is made of Syndrome on the official Pixar website implying that the others were born with them.
Syndrome wasn’t born with superpowers, but envied the attention and admiration the Supers received. Of all the Supers, Mr. Incredible embodied everything Syndrome desired: strength, fame, and popularity. Highly intelligent and cunning, Syndrome now spends his days on a remote island on the other side of the world inventing gadgets and robots to carry out a diabolical plot.
Pixar, The Incredibles, Syndrome