Salir a bolsa is an idiomatic expression, that is, a fixed phrase with a specific meaning that is not totally obvious from the words (like its equivalent in English, going public, I guess). It is a well-known phrase in the terminology of financial markets:
Salir a bolsa consiste en un proceso que se orienta a que las acciones de una empresa coticen en la bolsa, es decir, que se puedan
comprar y vender de manera libre. Esta acción permite a que personas
con ahorros o capital puedan invertir en una empresa para obtener
beneficios a medio y largo plazo, aunque siempre teniendo en cuenta el
riesgo que supone, y consiste en una acción que permite a la empresa
obtener la financiación que necesita.
Although the meaning of the phrase itself is not immediately apparent, the preposition a makes sense: together with salir it refers to movement "out into" some larger place, often with some purpose. There are many phrases that use salir a:
- salir a jugar = "go out to play"
- salir a hablar = "start to talk"
- salir al campo = "go out into the field"
- salir al sol = "go outside, under the sun"
There are instances where salir de means "to go out and engage in some activity", most of which refer to partying:
- salir de farra/parranda/joda = "go out partying"
- salir de compras = "go shopping"
- salir de paseo = "go take a walk/ride"
- salir de gira = "go on a tour"
Because of this usage, where the bare noun following salir de refers to certain kinds of activities, it's not likely that salir de bolsa would be understood as "go out and offer shares; go public". Instead it would probably be parsed in the same way as the common uses of salir de = "go out from, abandon, exit", that is, the exact opposite of the intended meaning.