While I was in Brazil, I heard that there is no such thing and it's against the Aikido principles, but I don't remember more than that. In the UK, I heard about some guys that do it. Is there really some fundamental reason that would make championships go against Aikido philosophy ?
It depends how one view "competition". If it is the desire to crush your enemies, see them driven before you, and hear the lamentation of their women ... Then, yeah, that style of competition is not compatible with Aikido. If it is a desire to learn effective techniques in a safe environment against opponents who vigorously resist so as to perfect your understanding of Aikido ... Then, yeah, they are compatible.
Aikido being the "Way of Harmony" has many political battles and camps. All of them quote O'Sensei as the source for them being right and everybody being wrong -- or charitably mislead. Ueshiba did spend a lot of time changing his own version of Aikido so quotes and insights while interesting should not be treated as canon.
Shodokan Aikido competitions formats are generally misunderstood: embu and randori are different beasts requiring different skills. Aikido Randori by Tetsuro Nariyama and Aikido and Randori: Reconciliation of Two Opposing Forces by Scott Allbright are goods book to pick up if one was interested in learning about competition with Aikido.
Most modern aikidoka reject competition and sparring in any form. The philosophy is delineated well on the Aikiweb forums by Stefan Stenudd:
[P]ractice is not about defeating an opponent, but about both participants being victorious by finding a truly peaceful solution and growing as human beings in the process.
We train the aikido techniques in order to perfect them, and thereby somehow also our minds. Non-competition is an attitude, not just the lack of fighting for medals, and should be trained as such. Only by diligent practice can we become non-competitive.
Tomiki Aikido (aka Shodokan Aikido), founded by Kenji Tomiki, is a notable exception. From the Aikido FAQ:
Tomiki Sensei believed that a "rationalization" of Aikido training, along the lines that Kano Sensei followed for Judo would make it more easily taught, particularly at the Japanese Universities. In addition, he believed that introducing an element of competition would serve to sharpen and focus the practice since it was no longer tested in real combat. This latter view was the cause of a split with O Sensei [meaning Morihei Ueshiba -Dave] who firmly believed that there was no place for competition in Aikido training.
It should be noted that the extreme restriction of techniques makes this form of competition significantly different from more liberal rule-sets such as judo.
Tomiki Aikido (aka Shodokan Aikido) holds tournaments and championships. There are competitive aspects in many branches of Aikido. There's quite little commonality between mainline aikido and the Tomiki schools. The idea was a "rationalization" of aikido by means of training in kata and competition, which Tomiki Kenji felt made the art more accessible (like judo, by which it was inspired).
Ueshiba Morihei however believed there was no place for competition within aikido. At this point, aikido was heavily influenced by the pacifist teachings of Onisaburo Deguchi, leader of the Omoto-kyou religion.