I am curious if it is illegal for someone who refuses to swear or affirm. I am wondering since a friend of mine tried to have someone to affirm and swear to tell the truth regarding an abuse investigation being done by my friend but the suspect refused to affirm and left the scene. Is this legal or not?
"Is this legal or not" is the wrong question. "How do I approach this problem in this context" is the right question.
By default a person has no obligation to cooperate with a generic investigator--why would they? There are ways around that in specific contexts and for specific professionals (failure to follow lawful order of a police officer, criminal charges for lying to a federal agent, cross-examination, discovery requests, etc...)
I am sure abuse investigations commonly have non-cooperative suspects and witnesses. Your friend needs to speak with experienced people in your friend's professional role who conduct abuse investigations in your jurisdiction and ask what their practice is when people are non-cooperative in this way.
I would refuse too - only a court or similar tribunal can require a statement under oath. Even statements to police are not under oath.
I personally would be extremely reluctant to say anything to an investigator who is that ignorant.