loads of the shops in Boro sell bumboards which are large round plastic trays with handles (i think there are 2 sizes available). My son got one last year and had hours of fun, we bent it into the top of the suitcase and will be taking it back in march.
My friend Sashor has a great shop in Borovets "BIg Foot" he is great for all your skiing needs he speaks fluent English, German, and being bulgarian is fluent in that as well. You can find him opposite samakov on the corner over the road from the Gondola station. I am pretty sure he will be able to source what ever you require. Have a great time
We stayed in the Rila in borovets last year, just opposite the entrance at bottom of nursery slopes, you can hire sledges for the children, my kids! (15 and 17) had a great time, they were dirt cheap, 5 lev for an hour or longer if no one else was waiting for them.