The problem with a phone is that the person has to have it on them and it has to be turned on.
I know Smartphones all have GPS capability, but most don't have an easy way for you to find the phone. iPhones have an app called Find My Phone that does allow for you to very easily locate the phone, using another iPhone or computer.
There is a program called Project Lifesaver present in most places in the US that includes the person wearing a device that can be used to track them and a specially trained group of searchers to find people who have wandered off.
http://www.projectlifesaver.org/
If you do a search using the words ALZHEIMER'S and TRACKING, you will find a number of other options, inlcuding things to place in shoes, clothing and in cars. Some are more discrete than others, but some don't work as well as others.
And, even though you don't want to take her car away, bad things can happen, even with early Alzheimer's. I think you are very smart to think of ways to track her.
This woman was from close to my home and I remember when she went missing. Her family was sure she could drive to her cabin because she did it almost every week. She had never gotten lost before, but had gotten confused in the Grocery store and 'turned around' a number of times.
She left her son's home in June 1999 and was never seen again - no trace of her car or anything in those 12+ years.