Don't know about sj but for eventing you need FEI registration for horse and rider and depending on the level of competition and a FEI passport for the horse then you enter via the particular event you want compete at. I would imagine a read at the BS rule book or a call to the office would tell what you need to know.
The details are normally on the entry forms for the show you are interested in. At a minimum you will need you FEI License number, your horse's FEI registration and FEI passport number. Remember you have to send your original horse passport to get the FEI one so you need to allow enough time, especially if you need to co-ordinate around travel or other competitions requiring passports. You can get more details here http://www.britishshowjumping.co.uk/page/FEI-Passports-And-Registrations
I've only ever competed in FEI shows abroad but the process will probably be the same.
You and your Horse will horse will need to be registered with FEI and your horse will need an FEI passport (For all 3 of my horses I had them back within 5 days of sending them off)
Entering shows isn't the same as entering national shows. Your federation (if you are British it will be BS) has to put the entries forward for you to the show. Honestly, give the internationals department at BS a call as they are so helpful.
Most shows normally have a quota of how many riders per nationality are allowed in so its best not too leave it too long for entries. The FEI schedules will have this quota written down somewhere!
Thanks all. I am already FEI registered from jumping abroad so will speak to the BS about it. Thought there might bee an easy online way i didn't know about.