Hello @KnowledgeSeeker and thank you for your question.
While the Copy Activity option to select Stored Proc may not exist for Oracle table dataset the same way it exists for SQL, there is something you can try.
In the Copy Activity "Use query" option where you can select "table" or "query", when you select "query" there is a space to write your own query statement. In this space you can write anything. Try writing an Execute stored proc statement here.
Please let me know if this solves your issue.