For the occasional one though they have a Minister at the venue (something else that's becoming more popular) and in these cases they are allowed to have hymns.
Finding organ hymn music (without choirs) is not that easy but I have found this site where MP3s can be purchased at 99p each - http://www.cul.co.uk/music/hymncds.htm
I actually first used them last year but lost the URL until now (I need them again & still have credit with them).
As with anything new there's a steep learning curve at the beginning. The 1st time I played one I found that the congregation were singing at a slower tempo so I had to slow it down (by 10%). The other thing to watch for is that you have enough verses of music. On the site mentioned the number of verses is given in brackets at the end of the title.
Apart from that there's nothing untoward that a DJ can't handle - just wait for the vicars cue and hit play, follow the hymn from the order of service and fade it at the end
