I am assuming you mean besides what the gym requires them to have?
I would think its the same type of items in any gymnasts bag regardless of what type of gymnastics they do.
DD does the standard type USAG JO program - Looking im my 14 yo's gym bag here is what I see
All the required items her gym wants her to have (comp leo, warmup suit, pre-wrap, tape to wrap ankles, wrists etc)
She also has her
grips and tiger paws with her for meets she brings from the gym.
For meets she also brings
*small lucky stuffed animal (new one each year) She ties on the ribbons she wins to it. This year its a frog last year it was a penguin. When the year is over that animal makes it to her award shelf wearing all the ribbons she won. Can't leave the house without it or its a major drama fest LOL.
*hair care products in case her hair falls out - extra hair ties, small travel size hairspray, hair pins, she wears her hair in a bun with a bun cover so she has an extra one of those, an extra hair scrunchy and of course spray hair glitter

how could she ever compete without it LOL, brush and comb (the comb with the long pointy handle)
Personal Hygine items - she carrys a travel size deoderant, toothbrush and toothpaste (with retainers lots of snacks can get stuck), Feminine hygine items, travel tissues, small first aid kit with the good sports bandaids.
Extra clothing - Extra leo in case something happens to comp leo like someone spills on her - it doesn't match the comp leo but it is in the same color scheme. Some parents actually buy 2 comp leos but at $190 one is all my DD is getting. She has a complete change of clothes in the bag as she arrives in her leo and warm up this way she can change after the meet.
Food and drinks - she always has some easy to eat snack items that doesn't need refrigeration and at least 1 bottle of water and 1 sports drink, if the meet is at breakfast, lunch or dinner time then a sandwich, bagle, muffin or something a little more filling may be brought.
Meds - she always has Motrin, some type of decongestant/antihistimine, so if she has pain for some reason she can take a motrin in the winter if she is congested she can take the antihystimine.
Pen and small note book - she like to write down her scores and any comments the judges make or coaches make on her performance.
gallon sized zip lock baggie to put trash into so it doesn't get all over everything