In the Cambridge dictionary "implore" is defined: showing in a very emotional way that you want someone to do something for you.
In this case Josephine asks Mrs. Mallard to open the door, she is very worried because fo all the emotions she is having. Josephine knows that her sister has a heart trouble and she thinks she is extremely sad fo Mr. Mallard's death. She desperately wants to enter the room to confort her sister as she could die form a heart attack.
As we can see, the definition above suites this case because of all the emotions shown by Josephine.