Yes, is the short answer. Riding competitions are an all weather sport, unless it is severe the show will go on.
We had a very rainny show day and the show still went on. It rained all day. Everyone will be wet by th end of the day. Raincoats are a great way to keep the rider and watchers dry. Umbrellas are a NO NO. They can and have spooked horses, which could lead to a an injury of horse, rider, or both. They make clear jackets to show in so you can see your number through and you can stay a little dry. I make sure I have my rain boots and coat available jsut in case it rains. I keep her show rain coat in our show box, just in case.
The course can get sloppy whether it has good drainage or not depending on how much it has rained. Cover your saddle if you are not on your horse. This will ensure you don't ruin it when in the rain. Make sure you dry off your horse the best you can just to help keep them clean. Also get any mud/sand off they may have flicked on them. This is a rain or shine sport. Don't worry about snow those shows are typically held inside if it is that cold. Just enjoy the experience and make the best of it.