This can be broken down in two parts. Can a jungler clear the jungle without smite, including objectives? Answer is yes, the jungle camps are pretty easy to clear. Warwick, Nunu, Tryndamere, Chogath, Trundle, Lee Sin, Nocturne, and other junglers who can top lane all have strong sustain to clear the jungle, albeit slower than if they had this summoner spell that did 400+ true damage to minions and neutral monsters.
The bigger question is should a jungler ever jungle without smite? This is a resounding no.
Without smite, the enemy jungler can counterjungle you so easily. They could just buy wards, and ward the brush by your red and blue buff. As soon as they see you start they can stand on the other side of the wall and smite with you being able to do nothing to stop them. Smite's damage at low level is higher than any other spell to neutral minions except ChoGath's Feast and Nunu's Consume. However these two abilities have animation associated to them that an experienced player can take advantage of and even use their abilities in conjunction with smite to outburst these two abilities.
This inability to secure buffs would continue to Dragons and Barons being freely taken by the team with smite.
The question now becomes, can the jungler have that much stronger gank presence with the summoner spell they took in place of smite to make up for all the free dragons, barons, buffs in the game. Considering, 1 dragon is equivalent to 3 kills and respawns every 6 minutes. The jungler would have to get 1 kill every 2 minutes as a direct result of having that different summoner than smite.
But this was a rhetorical question. At least one player on each side should have smite, this tends to be the jungler. So a jungler without smite is simply trolling their team to be at a disadvantage.
There are two last tidbits of information:
Having a spec in the mastery for Smite grants the jungler 10 gold each time they use it. This is a little bit of gold that can help the jungler a little as they can cast smite about once per minute. Finally, Smite can be cast even if you are stunned. In other words, if you are the jungler and in range to smite, the only way you can stop them from smite stealing the objective is to kill the jungler. On the other hand, if a Nunu or Chogath is silenced, suppressed, snared slightly out of range, knock backed, knocked up, feared, any kind of CC except a slow, they cannot use their consume or feast abilities.