Very nice were having this festivity yearly, but when the tradition started? Because it did not right after Saint Stephen's canonization.
The first who categorized it as a national - "official state" - holiday was Queen Maria Theresia in 1771 when the pope, in order to reduce the number of religious holidays, took out Saint Stephen's feast from the list of those. In order to celebrate the patron saint of her apostolic kingdom, Maria Theresa also ordered the Holy Dexter to be taken to Buda - from Ragusa (today: Dubrovnik), with a stop in Wien - it was presented to the Saint Sigismund chapel in the Castle of Buda.
Since her legitimacy was on wobbly legs, essentially in all her countries, she made an effort to cater to traditions and formalities, to prove and show she has all the rights she needs to rule over her domains. In Hungary too. Related to the topic on hand she also used the title of "apostolic king/queen" of Hungary, which technically all the kings of Hungary were entitled to use (very few did in practice) since Saint Stephen gained the right to carry the apostolic cross in front of him. She had a minor disagreement with the pope Clement XIII over this, whom questioned this entitlement, had a bit of exchanges of mail, and in the end he authorized the use.