Yes you can replace your own serpentine belt.
Replacing a serpentine belt yourself can be hard if it's your first time but if you know the serpentine belt routing which should be under your hood or at least online then it can be pretty easy to replace your old serpentine belt with a new one.
If you can take a picture of how the serpentine belt is routed.
Some serpentine belts are hard to get too and in those cases you might want to have a mechanic do it.
Also you should get a serpentine belt removal tool which helps tremendously when replacing serpentine belts because you'll need to release the tension of the belt by moving the belt tension pulley.
It may take you an hour to replace a serpentine belt if it's your first time and a shop should be able to replace the serpentine belt within 30 to 40 minutes depending on if it's easy to get too or not.
If you hire a shop or a mobile mechanic to replace your serpentine belt and pulley then the labor cost is between $50.00 to $150.00
And for parts the new serpentine belt is between $20.00 to $100.00 depending on the make and model of vehicle as some serpentine belts and pulleys cost more than others.
A new serpentine belt pulley is around $20.00 to $50.00 depending on the make and model of the vehicle.
If you replace the pulley and serpentine belt yourself then it will only cost you your time and the price of the parts.
A new serpentine belt for my car is only $50.00 and the pulley is $20.00
So for me the cost to replace the pulley and serpentine belt is $70.00 and I buy most parts online when possible.
A cars engine will still run without the serpentine belt.
However when the serpentine belt breaks or is not on the cars engine then the alternator will not charge the battery or produce voltage, the water pump will not work and you'll lose your power steering as well.
So you should not drive without a serpentine belt unless you need to get off the road and to somewhere safe.
I have had to drive a short distance such as 2 miles without a serpentine belt but only to get to a parts store that was close by.
Driving the car too far without the serpentine belt will make steering difficult as well as cause your engine to overheat and cause engine failure.
So you need to get the serpentine belt replaced before driving it too far.
Replacing a serpentine belt is usually pretty easy to do but some cars the serpentine belt is in a tight space and it takes some routing to get the serpentine belt installed properly and they can be confusing but you car should have a belt routing guide but if not it can usually be found online.
A new serpentine belt is between $20.00 to $100.00 depending on the make and model of the vehicle and an engine is more costly.