Yes, the Holy Sonnets could successfully be translated to music. They are quite regular from a metrical point of view and feature many rhymes and assonances, two important factors to consider when adapting lyrics to music. They are regular in form and have a natural rhythm which is also useful in order to determine where the musical accents go. They are also not very long, which is ideal for short pieces of music.
It would be possible to adapt these sonnets to modern harmonies, such as those we hear every day in pop music. However, as the Holy Sonnets were published in 1633, an interesting experiment would be to write the music in baroque style, which developed in the 1600s.
1633 would correspond the beginning of the middle baroque period. Music was evolving and certain characteristics typical of Renaissance music were left behind. New styles of composition were also developed. In this period, certain new vocal styles emerged, such as the opera, the oratorio, and the cantata. These typically featured short but effective melodies accompanied by simple harmonies and a bass line which integrated well with the melodies.