Strictly speaking, exposition is delivered by a narrator as part of fiction. In a play, there is rarely full exposition. That said, the Messenger who announces the play at the beginning does in fact directly address the audience, telling us what the play will be about and setting the stage for later action. That's the most direct example of exposition. However, given the allegorical nature of the play, you could argue that the over-elaborate explanations given by the other characters contain barely hidden exposition—what's sometimes called an "expository lump" of dialogue. Look at God's opening lines, for example:
I perceive, here in my majesty,
How that all creatures be to me unkind,
Living, without fear, in worldly prosperity.
Those are being delivered for us.
The exposition of "Everyman" opens with the Messenger announcing the play's purpose: "..a moral play./That of our lives and ending shows/How transitory we be all day." The Messenger tells the audience that Everyman is to be called by God to a reckoning and look to the end of his life as he begins it.
After God reflects upon man's evil ways, He then summons Death and instructs him to take Everyman on a journey. Protesting, Everyman offers Death gold and silver, but Death replies that wealth means nothing to him telling Everyman that everything he possesses "were but lent thee...as soon as thou art gone,/Another awhile shall have it..." Death then parts, instructing Everyman to make himself ready for the journey through life. Everyman feels bereft and worries that he has no one to keep him company on the forthcoming journey.