As a step-parent what legal rights do I have with respect to my step-children (attending doctors appointments, school conferences, etc)?

It is generally assumed that without adoption your legal rights to administer to the needs of your step children do not exist.  So participating in the the medical consultations when he natural parent is present  - even if the minor child wants you there - may not be possible if the natural parent objects.  You would have no legal right to decide medical care for the child.

However, if you are the "parent" home with the child when they require medication or emergency treatment a different scenario may arise. It may then become can the natural parent that you are married to delegate the duty to you in his/her absence?   Moreover, in an emergency situation do you have an obligation to act just as you would for a child who is a visitor in your house?  In an emergency I would say there is an obligation to act but you should notify the natural parent(s) immediately. The hospital would need them there to give consent for medical treatment.

The above is a general statement and not based upon familiarity with Vermont Law.  I would like to note that Vermont is one of the few states that imposes an financial obligation upon step-parents to support step-children. Therefore, I would check with an attorney in your area to see if there is a statute for standard of care that you can rely upon for administering medications, emergencies, etc. Good luck. Vt. Stat. Ann. tit. 15, 296 (1989) (stepparent has duty of support of stepchild).