The air conditioner doesn't turn on. If I turn the knob over to the vent, it gets warm, but the fan isn't running at all

it would appear that you will have to replace all items damaged during the theft attempt
as all the items are part of the dash assembly including the radio, I would consider a complete dash from the wreckers and that way you only have to hook up the wiring harness connections

May be in the radio installation. I hate pointing fingers at the last guy who worked on stuff, but they could have hit a blend door.

When car is warmed up, turn heat on vent and like 3/4 power for fan. Turn temp knob from hot to cold. You are looking for a temperature change. Now, turn on a/c to vent. Wait for it to blow cold. Turn temp knob to hot. Looking for air to be warmer that it was. POST RESULTS

You have broken vent flap servos or just need calibration with Volvo diagnostic tool.

Check whether the warm air isn't going in with the cold air .Coolant line that runs into the firewall on the passenger side of the engine which is toward the bottom half of the firewall on the passenger side. If there is warm air going into the vents with the cold air then it blows warm air. You may have to shut it manually because it gets stuck sometime. Check it out.

the fan is called a " blower motor" sounds like the motor is failing, try replacing the blower motor, there normally located under the passengers side below the dash

if the fan does not increase speed as turned up replace resistor assembly usually in heater housing under dash.if doesn,t nave separate resistor assy.it is the fan switch

thereis an actuator controling the flap t may be cracked causing the problem

If only the AC indicator light is on but you are still getting warm air from the vents, this is not an issue.  Most newer cars run the AC compressor nearly all the time to prevent moisture accumulation inside the car.  Without the compressor running, your windows may steam up and hinder your visibility.

Did the switch break because it was stuck? If so, then yes, they will have to change actuators and doors. Does the fan work on all speed setting? If so then you do not need a blower resistor. The resistor pack only controls the current flow to change speeds of the fan.