Yes, you can say: "Since I was a child, I have always wanted to follow in my dad's footsteps, becoming a chemical engineer." This sentence is grammatically correct and conveys the idea that you have had the desire to become a chemical engineer since you were a child. Regarding the use of "when" instead of "since," you can certainly use "when," but the meaning of the sentence will change slightly. "When I was a child, I always wanted to follow in my dad's footsteps, becoming a chemical engineer." In this version, the emphasis is on the period of time during your childhood when you had the desire to become a chemical engineer. The use of "when" indicates a specific point or period in the past when this aspiration existed. It doesn't necessarily imply that you still have the same desire now. So, both versions are grammatically correct, but they slightly differ in the emphasis and how they convey the timeline of your desire to become a chemical engineer.

Your sentence is awkward because the role of the clause "becoming a chemical engineer" is not clear. What is it? Is it an adjective clause? adverb clause? What does it modify? Does it modify "I"? Does it modify "wanted"? Does it modify "follow"? Does it modify "footsteps"? I can't figure out how to interpret it. The sentence can be fixed in numerous ways. You can replace "becoming a" with "and become" or "by becoming a" or "as a" and eliminate the comma. You can use "when" instead of "since" but then you must change "I have always" to "I always".