Most of the shows here in OK use asphalt for flooring which can be kind of tough on the horse's legs when they stand for a long time. I have small mats that are easy to haul and I put them down in the stall before I bed down. I always bring at least 5 bags of bedding in, depending on how many I have to buy from the show vendor. I bed deep and then I also have an extra stall as a tack stall/ready room with more mats on the floor, but no bedding, for grooming and prepping the horse to go in the ring.
I never leave expensive tack in the tack stall, my saddle & bridle and reins stay with me in the hotel room.
I just use the halter and lead rope as the extra chain on the stall, in 40 years of showing I've never had an escapee.
I bring a hay net that I keep full of hay to keep them occupied, 2 5 gal. Water buckets in summer, 1 in winter, and a bucket for grain & concentrates. Extra feed, salt, any supplements, ALL my grooming supplies.
I bring grubby clothes for me, show clothes and a coverall for once I'm dressed if I'm doing my own grooming.
In summer I use a cotton sheet and slinky, in winter I use a show sheet, heavy blanket, hood and slinky (for clipped horses only). I keep a small cart in my trailer for loading and unloading my stuff, a manure bucket & fork and my stall drapes.
I'm sure there's more but it's what I can think of right off.