it depends what you want to investigate. Bacteria which are able to colonize a kidney stone are usually relative robust bacteria due to the surrounding environment. If you want to make cultural analysis you should avoid desiccation of the material, which might kill a substantial number of bacteria (especially the Gram-negatives). With a low amount of physiological sodium chloride solution you will be able to maintain most bacteria for several hours. I do not believe that a long term storage will be useful for culture techniques. The storage at -80°C in a low nutrient medium containing 30 % glycerol might preserve some bacteria. But I even expect a substantial loss.
If you want to go in the direction of metagenomics all storage conditions suitable for long term conservation of DNA and/or RNA would be sufficient.