Can my mom buy an auto insurance policy for her car in the states without being in the states? e.g, fax, sign digitally, etc.?

Yes. If she is a resident of the states and her car is in the state. Insurance companies don't require a signature....just payment

I never had to sign anything for my insurance policy. As long as you plan to pay them, insurance companies would LOVE to sell you an insurance policy.

Yes. It is usually possibly to buy insurance over the phone or online from Geico or Progressive.

However, the price is based on the ages of the drivers. If she cannot drive it because she is not where the car is, so you will have to be the only driver, then she will have to pay the price for a car driven only by a person your age, instead of the price for a car driven by someone her age.

She must be resident in the system the car is registered in.

I have never had to sign insurance policy form. I just called my agent or went online.
So I would think she can get insurance without any problem.
If she is just is going to be back in a few weeks, no big deal just get the policy.

Here is the tricky part. Does she live out of the US, meaning her permanent residence is outside the US?
And this issue would apply if she lived ANYWHERE other than where the car is located. 1 mile, 10 miles or 1000 miles, the same issue would apply, when you get insurance you are stating that you have control over the vehicle. So really what she is saying is that I am getting insurance, but I can not monitor the vehicle and others are driving it. I am not sure what the insurance company would say here, you would need to ask them, and you would NEED to send in a letter stating your situation and send it via certified mail and keep copies of the letter and mail receipt in case you get into an accident and they say, Sorry not paying your claim the car's owner does not live at the policy address.
Other option is to re-title the car as joint or single person title to the person who will actually be driving the car.
Rules vary by state. Some states may allow a power of attorney form , might need to be notarized. Some require the title to be notarized. Check the DMV web site for details.
Or ask a local attorney if you are not sure.

She can, but you're leaving out some really important details. Like why she isn't in the States but owns a vehicle there.

No. She must reside at the address where the insurance is registered.

No sir. Your mom can only insure, register and title a vehicle within the state she is a permanent resident. Motor vehicle insurance policy cost and coverage is based on your specific area of residency.