If Long Day's Journey has a protagonist, it's Mary. She seems to be the main focus of the play. O'Neill ends each act with her and makes her final hallucination the grand finale. Of course, in drama, a protagonist, is most basically defined as the character that causes everything to happen. Does Mary fit this description? To a certain extent.
Mary's choice to start using morphine again is the sorrow around which all other sorrows revolve. The play seems to be made up of a series of reactions to her slow descent. Even when the other Tyrones aren't talking about Mary, the tension caused by her renewed addiction lies just beneath the surface. Edmund probably puts it best when he says, "She moves above and beyond us, a ghost haunting the past, and here we sit pretending to forget" (4.1.148).
The argument that Mary is the protagonist is strengthened by the fact that she's the only person in the play who is looking for something. Of course, she never seems to be quite sure what it is. Our best guess is that it's some combination of the things she's lost over the years – innocence, purity, friends, a baby, happiness. Unfortunately, Mary is unable to find any of these things. Instead, she just sinks further and further into the haze of morphine. (Check out "What's Up with the Ending?" and Mary's "Character Analysis" for more.)
Even though we think there is some room to argue that Mary is the protagonist, we should point out that most folks agree the play just doesn't have one protagonist. While Mary does seem to be the focus, she doesn't do quite enough for some critics to dub her the protagonist. Basically, she takes morphine, blames everybody else, and takes another hit. If she is a protagonist she's an inactive one.
None of the other Tyrones do much, either. There may be some small case to be made for Edmund. He's the most blameless of the Tyrones, and could be seen as a window for the audience. The fact of the matter is, though, that he does very little. Mostly he just sits around drinking and getting bad news. He's one of the most passive characters in the play, making him an unlikely candidate for protagonist. Jamie and James are equally as inactive.
The argument could be made that (though Mary is the closest) there's no one character that drives the action of the play. Long Day's Journey doesn't really have a larger overarching plot. It's really just a series of smaller repetitive actions. The Tyrones hurt each other then forgive. It's the same thing over and over again. The play shows a family caught in a vicious cycle of hate and love. Perhaps, rather than saying that nobody is the protagonist, we should say that everybody is.