Who's asking: Anna Bragdon, China, ME
My knee-jerk response to this question was, "Oh, sure" -- but when I went looking for infants in my Collected Shakespeare, they were pretty hard to find. It makes sense, I suppose, since even women didn't perform on stage on Shakespeare's time.
As far as I can tell, the only Shakespeare play that calls for a baby on stage is The Winter's Tale (Act II, Scene iii). Paulina enters with the baby Perdita, who is banished from her father's court because her father, King Leontes, thinks she's someone else's child. Shakespeare mentions the baby Queen Elizabeth, though not by name, at the end of Henry VIII, but we don't see her.
The Winter's Tale also has a speaking role for a child (Perdita's older brother, Mamillius, in Act II, Scene i). The only other plays with roles for young children seem to be Richard III (with a girl and boy who are children of the murdered Duke of Clarence and the doomed young princes who are sons of King Edward) and Macbeth, in which Macduff's son has lines.
I consulted my old friend John Erath, who knows much more Shakespeare than I do, and he came up with a few more possibilities. The "groaning Juliet" in Measure for Measure could have a baby in arms at the end of the play; Titania in A Midsummer Night's Dream has a "lovely boy, stolen from an Indian king," who could be a baby. Shakespeare's plays often have crowd scenes, so if a proud parent wanted a child to make an early Shakespearian debut, it could certainly be as a carried-on extra.
It's entirely possible that I've missed something, so if you can think of any other small children in Shakespeare's plays, post it below.
To continue the theatrical theme, I invite everyone in the Augusta area to come out for Gaslight Theater's production of The Mousetrap, which opens tonight. Performances run tonight and next Thursday at 7:30, this weekend and next weekend at 8:00 p.m.