This question does not belong to history of sciences and math in the strict sense. This is a question about philosophy of mathematics and it has nothing to do with convolution. Convolution is just an example of a mathematical object.
You can ask about any mathematical object whether it was invented or discovered.
And the answer depends on the philosophical views of the person who answers.
Therefore, there cannot be an objective answer to such a question.
Personally, like most mathematicians, I am a Platonist, and I believe that mathematics exists independently of us, and we discover it.
Some others ridicule this point of view. But there is no way to reach a consensus, like on
any philosophical question. So this question has no answer.
Convolution is as "natural" as addition or multiplication. So if addition and multiplication (say, of integers) "exists" in the real world, and we discover it, then
the same can be said about convolution.
But of course there are people who will say that addition and multiplication were invented. (Did not exist before humans started to add and multiply). And history shows that there is no reconciliation between these two sets of people.
EDIT. The answer of Danu which was simultaneous with my answer gives an example how convolution arises in probability theory. But convolution arises everywhere. If you think for example of multiplication of decimal numbers, this is nothing but a special case of convolution.
As I said, from my point of view all this "exists" and we discover it. But some people (mostly philosophers) will say that we "invented" probability and multiplication, and they
"did not exist" before we invented them:-)