This couplet means "red plums in bud are proud of winter snow, and green willows spit catkins to welcome the new year."

Plum blossoms are in bud in winter snow: Plum blossoms are in bud in cold winter (indicating that spring is coming).

Green willows spit catkins to welcome the new year: willows spit buds to welcome the arrival of spring.

There is something wrong with "green willows vomit": all green willows vomit in late spring. It should be "green willows sprout to welcome the new year"