The Shodokan/Tomiki school of Aikido holds competitions regularly across the world. They have done since the 1950's so far as I am aware. Kenji Tomiki was a student of Morihei Ueshiba and Jigoro Kano, and was promoted to 8th Dan in both Judo and Aikido. His Judo influenced his ideas on Aikido, which eventually led to him formulating a form of free fighting, which eventually lead to competitions.
Like Judo, there are varying levels of score (e.g. waza-ari, ippon) depending on how well the technique is performed. Generally, for throwing techniques, scores are awarded if your opponent lands on their back/side with significant momentum while you maintain reasonable balance. Scores are awarded for locking techniques if you apply a lock fully and can maintain the lock for a few seconds, leaving your opponent "defenseless".
There are 2 types of Randori: Tanto randori (where uke has a small wooden tanto and attempts to hit tori with it) and toshu randori (empty hand randori, not a million miles away from pushing hands in Taijiquan).
Generally speaking, the throws/locks applied randori are one of the 17 kata of Shodokan/Tomiki Aikido:
Hope that helps! Anymore questions, fire away
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