Is this normal in a nursing home?

What sort of nursing home is it, Larryss? In a memory care or dementia care unit it wouldn't be so surprising - perhaps a bit more surprising that the strangers allow her to do it? - but if it's supposed to be a standard facility for elder care then perhaps it's time for this lady's family to think about stepping up the level of support she's getting.

Half or more of the residents in most nursing homes have dementia. So unusual behavior can be expected. The staff should be aware of this behavior and take what steps they can to prevent it. And the strangers certainly don't have to permit this. If this is the resident's only inappropriate behavior perhaps the staff and/or the family doesn't think she needs a higher level of care yet. Or perhaps transferring her is in the works.

You might want to talk to the director of nursing to learn what is going on -- for example what steps are they taking to try to prevent this? Of course the DON will be limited by confidentiality concerns, but may be able to answer you in general terms.

I think you can always expect the unexpected in these situations. My own mother frequently touches people, usually on the arm but sometimes will touch someone's face! And it doesn't matter how many times you tell her not to, she will unwittingly do it again.

Had one room mate in the hospital who constantly came over and patted me and rearranged my bedclothes to "tuck me in" Fortunately removed in the middle of the night because they needed the bed for a young mentally compromised woman drug adict who had been consuming alcohol. I spent the rest of the night using my call bell to let the nurses know she had her legs through the bed rails AGAIN. Fortunately she was transferred the next morning.
So yes expect the unexpected in any facility. Just don't tolerate it.