If there aren't published vectors to tell a commercial pilot which heading to take as they climb, can they decide where to make their initial turn on their own or would an ATC always involved in where they make the turn?
The short answer is yes, a commercial pilot can make a decision when to turn.
To clarify a few points though, there really isn't such a thing as a "published vector". If a departure procedure with heading were to be published it would then become a Standard Instrument Departure, or SID. True, some SIDs will specify radar vectors, but this simple tells pilots that they should expect ATC to assign headings that will vary depending on traffic, not which heading they can expect to actually fly.
Next, "commercial" pilots don't do anything different than private pilots, airline transport pilots, or military pilots. When and where you turn during departure depends primarily on whether you are operating under IFR or VFR. Secondarily, decisions on when to turn will be based on terrain, noise abatement, company policy, and lastly personal preference.
So, considering my answer in light of the above, there is a big difference between a 737 operating IFR on a SID, and a commercial pilot in a 172 towing a banner under VFR out of a non-towered airfield. If you have a specific scenario you are wondering about you might get a better answer if you can narrow down the question.
It depends. Commercial pilot could be doing a VFR part 91 flight from his own backyard.
I suspect you meant pilot of an airline flight? Yes, in one way or another he will be told which way to turn 99.9 % of the time, in the USA at least. Whether by tower, ODP, or company guidance.