Belle is an Irish Setter, a beloved member of the Keller family. Although she is not a character usually focused upon in The Miracle Worker, I applaud you for asking this question in order to highlight such a special minor character here!
Let's take a look at one of the first instances when we see Belle with Helen (apart from when Belle is simply sleeping by the pump). In the stage directions on page 31 it states:
In the house the setter Belle flees to the family room, pursued by Helen with groping hands; the dog doubles back out the same door, ...
From this, we can infer that Helen gives affection (probably very rough affection) to Belle whenever Belle is willing to take it. In fact, this is confirmed when we see Helen interact with Belle near the end of the two-week hiatus from the family:
When [Helen] encounter's Belle, she throws her arms around the dog's neck in delight.
Kate: Doesn't she need affection too, Miss Annie?
Now, these two instances are fine and good, but the instance that I think you are referring to is a bit later in that very scene around page 99. Annie is desperately trying to explain to the Kellers why she needs more time with Helen and exactly why their affection for her could be to her detrement. In the middle of the conversation, Helen gets their attention:
Helen is playing with Belle's claws; she makes letters with her fingers, shows them to Belle, waits with her palm, then manipulates the dog's claws. ...
Keller: Teaching a dog to spell
The dog doesn't know what she means, any more than she knows what you mean, Miss sullivan. I think you ask too much, of her and yourself. God may not have meant Helen to have the--eyes you speak of.
Perhaps more important is how Annie manipulates the situation, gently chiding Helen:
Annie [Gently]: No.
(She shakes her head, with Helen's hand to her face, then spells.)
Dog. D,o,g. Dog.
(She touches Helen's hand to Belle. Helen dutifully pats the dog's head, and resumes spelling to its paw.)
At this point Annie gets some water for Helen, trying to make her understand. However, Helen continues to try to spell water to the dog and even tries to "thrust Belle's paw into it."
It is safe to assume, then, that although Belle become's part of Helen's instruction, Belle is not the instrument of enlightenment as the water from the pump is. Just as the family (too in love with Helen herself to properly help) can never be Annie (the true teacher). Belle (the beloved family pet) can never live up to the sensory experience of water (the item burned into her memory from an early age through sight, sound, touch, taste, and smell): the only word Helen had ever spoken, ... and the first word she will speak again.
At one point in the story, Helen tried to finger spell into her dog's paw. Belle was her dog. She was excited about learning to communicate. She just wanted to share her new found knowledge with Belle:
Helen Keller had a lifelong love of dogs and always had at least one or two by her side. When she was a young girl, she even tried finger spelling, her only way of communicating, into her dogs' paws.
Of course, Helen was just coming out of her dark world. Light was breaking forth in her mind. She just assumed everyone used sign language. She could not comprehend that people and animals comunicated differently.
Helen had so much to learn, but at least she now had a way to communicate. As a child, Helen did not yet understand that dogs and humans communicated differently. From the time she was nineteen months old, she had never heard anyone speak. She had never heard her dog bark. She had never seen anyone communicate. She lived in a dark world with no communication. Once she learned to communicate, she wanted to share it with Belle because Belle was her companion.