What is the ruling on video recording the Friday sermon with a camera for the purpose of entering a competition among imams?

Taking pictures with a camera is an area of differing opinions regarding its permissibility, and it is more prudent to avoid it unless there is a valid need. The purpose mentioned in the question (recording Friday sermons) contains a benefit that justifies undertaking this action. It serves as a reason for the preacher to strive for an excellent sermon and as a means to select competent preachers and place them in appropriate locations.