Not necessarily. This can fall under the fair use doctrine, certainly as to the song recording and the submission of the song. This gets tricky in that the "artist" has to pay to enter their song and was paid for winning on a website that presumably published or made available the performance. There is certainly an argument that this then violates the fair use doctrine by the website owner, if nothing else, if they did not secure a license.
It, in some respects, is no different that a bar purchasing a generic rights license so that a cover band can play songs at their bar.
If I were advising the website owner, I would certainly advise them to purchase a license from BMG, or one of the licensing agencies, to cover myself against an infringement law suit. I think this falls into a gray area, but it is close enough that I think a license would be needed by the website owner to avoid any potential issues with infringement and infringement litigation.
Music copyright owners are really cracking down and suing on even marginal stuff right now for a number of reasons that don't really matter as it pertains to your question.
Point being, the singer/artist is probably okay here...but the website owner is likely infringing the copyrights of those that are artists whose original works are shown on their site.
Please rate my answer positively and let me know if you have any other questions.
Best wishes going forward!