No. There should not be any music during the sign of peace.

I think it's rather silly, in the context of the OF. For one, the sign of peace should be a brief, sober rite, according to the "traditional practice of the roman rite" (GIRM no. 42).

But if (a big IF) you take for granted that the sign of peace will oddly be extended, looking at the context of the EF where organ improvisation frequently happened over other actions, I think you could actually make a decent case for it.

Not trying to stifle or censor further discussion, but why is this question necessary?

Happily, at my parish the sign of piece is a brief rite: nobody wanders around, there's no chatting, the sign of peace is only exchanged with people in one's immediate vicinity. After less then a minute or so, the choir starts singing the Agnus Dei right away, which demands the participation of the people, and the sign of peace comes quickly to an end.

Why is this question necessary? The Rite of Peace should be brief and sober. Brief, because the sign of peace is only to be offered to those who are nearest. Sober, because the liturgical law asks for it. GIRM 82 says:

It is appropriate that each person, in a sober manner, offer the sign of peace only to those who are nearest.

Sacramentum Caritatis 49 confirms:

Even so, during the Synod of Bishops there was discussion about the appropriateness of greater restraint in this gesture, which can be exaggerated and cause a certain distraction in the assembly just before the reception of Communion. It should be kept in mind that nothing is lost when the sign of peace is marked by a sobriety which preserves the proper spirit of the celebration, as, for example, when it is restricted to one's immediate neighbours.

Organ improvisation covering the sign of peace doesn't communicate this brief and sober character which is proper to the rite. Contrary, it confirms and maybe encourages an exaggeration of the rite, and might even add to the ‘distraction in the assembly’.

Once I see the priest returning to the altar for the Fraction I begin the intro to the Agnus, this curtails the Pax and we can move on; I've seen some places where the priest goes back to the altar and everyone just waits until the Pax is over (5 minutes some times) and then the Fraction and Agnus are begun.

Use the Agnus Intro to put an End to Peace!

"There should not be any ... sign of peace." FIFY, ContraB.
Seriously... an improvisation could prolong the Pax. If you give folks enough time to greet their most immediate, and then start the Agnus (or an intro to the Agnus), they'll get the message.

Under normal circumstances, I don't play any music during the sign of peace because it is supposed to be brief. I am also watching the priest intently so as to begin the Agnus Dei on time. Nevertheless, depending upon the priest, sometimes the sign of peace is extended. Some celebrants actually leave the sanctuary to exchange the sign of peace with the congregation. In those cases, I will improvise on the introduction to the Agnus Dei until the priest returns to the sanctuary.

My first master, immediately after "Let us offer each other the sign of peace", would draw the Pedal 16' Subbass and Swell 8' Viola (box nearly closed) and maintain a gentle, "meditative", continuous improvisation all the way through the "Lord I Am Not Worthy" to the beginning of the Communion chant. If there was no choral singing during Communion, he would then return to the quiet, meditative mode until the closing of the Tabernacle. (Other quiet registrations were used at times, "Vaporizing on the Voix Celeste", etc.)

I must say that it DEFINITELY added to the quiet, contemplative atmosphere, and did not allow for any cacophonous "chattiness" to emerge during Communion.

Gaudete in Domino Semper!