Can I request songs?
Whether you can request specific songs or tunes depends on a number of factors:
String Quartets and Similar
If the piece of music is already in the quartet's repertoire, then probably you can request it provided you do so well in advance. Bear in mind that some pieces of music may not be played very often, and will need some rehearsal. Quartets are busy working and often can't get to rehearse easily. So make sure you give them plenty of warning of what you want.
Another issue is pop music for quartet. People often ask for something they have seen a quartet playing on YouTube and think that any quartet can play it. There are a few quartets around the world who specialise in doing their own arrangements and recordings of pop [often with backing rhythm sections], so it doesn't mean this can be played by most quartets.
If the quartet have an arrangement of the piece of music, then no problem. If one of the music arrangers and publishers who produce music for quartet has done an arrangement, they may be able to buy it. Other quartets will arrange certain pieces, but usually for an additional fee. Again if they need plenty of warning of this. Often pieces of pop music are not suitable for a quartet, or the version you saw on YouTube included a full backing group and rhythm section to make it sound as it did.
If you must have a particular piece of music, you must specify this at the time of booking.
Jazz ensembles are generally well equipped to do particular requests, provided it is something for the style of jazz they play. If you must have a particular piece of music, you must specify this at the time of booking.
Pop bands of various kinds
Pop bands tend to have prearranged sets. If what you are asking for is in one of their sets, that is OK, but these kinds of band are generally less flexible in accommodating requests outside their existing repertoire. If you must have a particular piece of music, you must specify this at the time of booking.
What if it turns out that the band can't play the piece, or the price for providing it is higher than you can accept
do the band play requests; I want a special tune; I want a special song; will the band play new songs; do the band do special arrangements; I want my own song; I want a special song; I want to choose a song; I want to choose the music; will the band play anything I want / ask