I don't believe so. ようになる is about something coming into being over time. Getting good at tennis, speaking Japanese, etc. つつある is about the actual process, not the finished product... if that makes sense.
つつある describes the process of getting to some point. ようになる is about the finished product or the idea of getting to a certain point. Examples of that again are getting good at tennis or being able to speak Japanese fluently. I know I'm not explaining it perfectly so I suggest checking the site below. ALC is always a great resource for sentence examples.
Looking at it from a historical perspective, this appears to be a holdover of the classical ~つつ ending, equivalent to the modern ~ている (c.f. 流れつつ、流れている). Probably trivia mostly, but thought I'd throw it out there.