Absolutely you are allowed! In fact, it is your obligation as a guardian of your son under Section 37 of the BC Family Law Act to make sure he is clean, groomed and well cared for.
Ideally, you would communicate with your ex about the need for a haircut and include them in the discussion about where to get the service and what type of style to request.
As your son gets older it will likely be his choice how his hair is styled and both of you will have to respect his choice. My clients often find that taking their children to a hair salon rather than doing the cut themselves helps to diffuse any criticism from the other parent.
Any guidance provided is not covered by solicitor–client privilege, nor is it taking into consideration all of the facts of your matter beyond those in the question. The legal information is specific to British Columbia Law. If you want a more thorough and case-specific analysis of your legal matter, please contact us to arrange a consultation.