I'm not familiar enough with that engine to say for sure, but the circled pulley might be the water pump. The wobbling is not a good sign, it indicates that a bearing is failing.
There are four accessories that are likely to be driven by a belt or belts at the front of the engine:
- The water pump (it will probably have a largish hose or hoses with hose clamps on them connected to it – or there may be nothing connected to it at all, some pumps work entirely within the block),
- The alternator (it will be the only one with big wires connected to it, it will also usually have an open frame and you'll be able to see more wires wound inside of it),
- The power steering pump (it will be near the top with a cap for checking and adding fluid either on the pump or close by, there will be two relatively small hoses and they will head towards the firewall of the engine), and
- The air conditioning compressor (it will have a couple of relatively small (compared to the alternator) wires connected to it as well as two hoses, the hoses will be small, similar to the power steering pump, and will not have hose clamps).
The final possibility is that that is either the tensioner or an idler on the serpentine belt. If either of those fail, you're likely to loose all of the accessories driven by the belt. The tensioner will be on an arm that allows it to move and keep the belt tight. Idlers will just be there to change the direction of the belt.
Most likely the problem is with the water pump. It would be risky to drive with it in the condition you describe, but you might make it. If the pump fails you may loose coolant or coolant circulation (or both). You'll probably first notice the loss of circulation as a loss of heat. If it starts leaking you may smell as sweet smell and notice the temperature going up. Carry extra coolant with you if the problem is with the water pump.
Another thing to be concerned about is the other accessories driven by the same belt. If the wobbling pulley fails you may loose the others as well.