Simply stated and for disclaimer’s sake, I do not have a firm statement on where this comes from.
My guess is that the ombrellino’s symbolism is related to the College of Cardinals. According to the old Catholic Encyclopedia (~1915), one of the rights that a cardinal enjoys is the use of the ombrellino which “is held over them whenever they quit their carriages to accompany with bare heads the Blessed Sacrament, if perchance they meet It on their way.”
Looking past that, an ombrellino, defined broadly, is a type of canopy. From what I gather, it differs from the baldachinum, or altar canopy, in that the baldachinum is made of stone, metal, etc and has multiple pillars. A form of these are/were allowed over the chairs of prelates and of princes. In this case, this is referring to secular princes, although cardinals are known as “princes of the church” so there may be some connection here as well. If you couldn’t tell, it’s all old medieval-style adornments.
The ombrellino is noted as being smaller and carried by a single staff (i.e. like a umbrella). Lastly, many years ago, the ombrellino was used during conclaves over each of the cardinals. If I remember my history correctly, when a new pope was elected, all the cardinals would pull on their ombrellino causing it to close with the new pope’s ombrellino remaining open.
In short, put that all together, it seems that the use of the ombrellino in the crest of the vacant see is to indicate a) the absence of the pope by contrast and b) the role of the cardinals in guiding the keys of St. Peter to the new pontiff.
Anyone have anything to add to this?