Can the 'Successor Agent' simply move forward and take charge as POA?

I look at POAs every day for work (financial services - all POAs that are questionable come to the legal team for my review). When we have a successor agent step in while the primary agent is still alive, we ask for the first agent's resignation. It doesn't have to be complicated, just that the primary is no longer able to be the agent. Some states may require the new agent to "accept" (it's usually labeled "Important Information for Agent"). If this isn't required in your state, it certainly doesn't hurt to have the successor agent execute a simple document indicating that they have accepted the role of agent in light of the primary agent's resignation. In a legal context, it's really not a big deal. Agents can become incapacitated, ill, or simply not want to act as POA. That's why people should name a successor agent (or two!).

If the POA was recorded for real estate purposes, it would be a good idea to re-record the POA with the resignation and acceptance, so that it is a public record as to who is actually acting as agent on your sister's behalf.

Best wishes.

That's why there is a successor agent named in the document -- so it can go into effect even if the principle is unable to name someone at that time.

While don't you call the office where the original was registered and ask if they need to be notified when the successor takes over?

Why not? This situation sounds exactly why a successor is set up in the first place.