Can I crank the engine with exhaust off the car?

Hello,

Is the exhaust manifold still on the car? Are you just missing the exhaust piping?
You can drive without the exhaust pipe, but it is not ideal, and you do not want to drive it for an extended period of time.

If you are missing the exhaust manifold, I would advise against driving it.

Let me know of any questions.
Thank you.

https://www.2carpros.com/articles/how-exhaust-systems-work

They can but it will be loud. It may set a couple of O2 sensor codes as well because they won't be working. Those should clear themselves as soon as it's back together though. I wouldn't drive it a lot though as it will also be dumping the exhaust right below the cabin and could cause bad things to happen to the driver, as well as getting a ticket or two if the police catch it. The shop may not want to drive it for those reasons though.