Daggers were not used in tournaments in the narrow sense of the usual medieval definition of tournament, that is, a group combat on horseback. In the broader sense that many modern writers use the term to describe a limited armored deed of arms by consent, they could be used.
Deeds of arms on foot can be divided into two categories. In one the fight ended when one side or the other had struck an agreed number of blows, either with a single weapon or with different weapons in turn. If multiple weapons were used there was a pause between weapons, so the additional weapons might be either held by an attendant or kept in the champion’s tent. Dagger was one of the weapons used, or intended to be used, at the following combats for an agreed number of blows.
Challenge by Michel D'Orris 1400
Richard Beauchamp vs. Pandolfo Malatesta 1408
Continge vs. de Bars 1415
In other combats there was no limit on the number of blows. In the most extreme form, the combat could continue until one sided was dead or surrendered, or the judge stopped the fight. In more limited combats the fight would stop as soon as one side was carried to the ground or disarmed. In these the champions would typically be armed with multiple weapons in case they lost their primary one: pollaxe, sword and dagger or pollaxe and dagger, and sometimes a spear as well.
The Seneschal of Hainault Performs a Deed of Arms in Valencia, 1403
Duke of Bourbon's Enterprise 1415
D’Ollumen vs. de la Haye 1415
Alvaro Continge vs. Clugnet de Brabant 1415
Three Portuguese Do Arms against Three French at Paris, 1415
A Combat between Sir John de Mello and the Lord de Chargny, 1435
Asteley vs. Boyle, January 30, 1442
How Sir Jacques de Lalaing did arms in Scotland; and of many other particulars in the house of Burgundy. (1449)