Camping is certainly possible in the park but you must register at the Cemoro Lawang gate (where there is an adjacent campsite). There are many sources of safe, fresh water in the park - ask locally.
Potential campers should be very aware of how cold it gets here though and be thoroughly prepared for that. Heavy duty sleeping bags are essential.
Mount Semeru erupting in 2004
Temperatures on Mount Bromo are refreshingly cool during the day (although sunburn is still a real danger), but outright cold at night, as temperatures can drop to zero in the dry season and are rarely much above 5°C in wet season. Some of the cheaper places to stay may not provide adequate blankets or heating, so come prepared. If needed, you can rent jackets and hats at Cemoro Lawang and at the Penanjakan viewpoint for about Rp 20,000. Hats and gloves are important if you wish to survive in the cold especially early in the morning, face masks would keep the dust away if the crowd gets big around the crater.
There are cases of malaria each year in the lower foothills of the park and any visitor planning a long stay or to camp in this area should take necessary precautions. This is not though a problem for those visiting Mount Bromo or the high plateau only.