There actually is not a spot in the play in which Romeo and Juliet say that they would be willing to die for each other if they were ever separated. However, in several of the lines they exchange with each other, they make allusions to death, foreshadowing for the reader that their deaths are soon to come.
We see the first allusion to death in their conversation in the famous balcony scene. Towards the end of this scene, Juliet compares Romeo to a pet bird and herself to the child who lets the bird "hop a little from her hand" only to drag it back again like a prisoner. Romeo responds that he wishes he were her pet bird, and Juliet makes the foreshadowing reply,
Sweet, so would I.
Yet I should kill thee with much cherishing. (II.ii.196-197)
This passage foreshadows their future deaths because, indirectly, Juliet does become partially responsible for Romeo's death.
A second allusion to death takes place when, after consummating their marriage, Romeo is climbing out of Juliet's window to leave for Mantua. As Juliet looks down on him in the garden from her window she sees how pale he looks in the morning light, and it gives her a foreboding feeling. We see this when she declares,
O God, I have an ill-divining soul!
Methinks I see thee, now thou art below,
As one dead in the bottom of a tomb.
Either my eyesight fails, or thou look'st pale. (III.v.53-56)
This passage again foreshadows their upcoming deaths by likening Romeo to a dead man in a tomb. Romeo even responds by saying that she looks dead as well and that it is due to their sorrow, as we see in his lines,
And trust me, love, in my eyes so do you.
Dry sorrow drinks our blood. Adieu, adieu! (57-58)
Hence, even though Romeo and Juliet do not make a vow to die for each other, their deaths are foreshadowed all throughout the play and even alluded to by Romeo and Juliet themselves. Therefore, it is not a surprise for the reader to learn that they both die at the end of the play.