I am in a shared parenting situation and a situation arose where our son needed to go the Dr. I was in training and the mother claims to be out of sick time. I was told by the mother that her mother (son's grandmother) would be able to take him.

Just yesterday the mother claimed that her boyfriend took our son the Dr. appointment and I would like to know if this violates any health privacy laws such as HIPAA.

Nothing has been violated as the mother gave permission.

Mother has the right to let ANYONE take our son to a dr appt? How is that not giving someone else the ability to make medically related decisions to our son's health?

Medical decisions? Depends on the reason for the appointment. I'm sure he has not been given the authority to make decisions, but mom was caught between a rock and a hard place and had to have the son seen by the doctor. The final decisions will be left up to the mother.

Mother has the right to let ANYONE take our son to a dr appt? How is that not giving someone else the ability to make medically related decisions to our son's health?

As J_9 said, you haven't indicated that any medical decision has been made by anyone. The BF simply gave the child a ride.

But, assuming that the BF did something more, and said, "Sure, Doc, do the procedure,", how would this be different than if Grandma had given the child the ride and authorized treatment?

Finally HIPAA has to do with disclosure of information, not treatment authorization.