This is an answer for which, at first view, one might think we'd have to drive our analogies rather too forcefully in order to draw a true connection. In The Glass Menagerie, Laura is a sheltered young woman, kept out of contact with others and disabled. Hester Prynne, by contrast, is the proverbial "fallen woman" of the old patriarchal system of judgment. Not only has she had a child out of wedlock, but from the very first, we can see it as a deliberate defiance of the system, in the (apparently) proud and elaborate way she had sewn the "A" on her dress.
The similarity between Laura and Hester, however, is that both are outcasts. A girl living at home, isolated and destined to become a "spinster," was judged and marginalized in a way not altogether differently from a woman who violated the sexual codes of those earlier times. Both women are the Other, in outwardly opposite ways which paradoxically merge into the same position unfairly judged by society.
Williams based the character of Laura to some extent on his own sister, who was diagnosed with a mental disorder. There is something eerily similar in the exclusion of Hester from Puritan society, as if "lapses" with regard to the moral code of that time were treated as a kind of mental illness. Both Hawthorne and Williams are making a statement about the especially unfortunate status of women who are judged for being different from what is expected of them.