As an owner of a Property Management company, a Contracting company, and rental owner I can weigh in here.
It is illegal, doing so involves civil penalties. All PM's carry E&O insurance to protect you. There is a real risk of being sued, whether your wrong or right. Anything you do when speaking to a tenant is pretty much is illegal when you ar managing.
Legally, if you have some ownership or are a w2 employee you can do this. Also, they could write a lease to you, then have you sublease to others. This will get you around the laws.
To do it correctly, you need a license and a broker. This means you pay the broker a percentage. You also pay annual fees, attend continued education classes, etc.
I would caution you however, that you should instead offer a different approach. What they want is, someone to take calls, do the maintenance, and fill the vacancies. They want you to take on life's everyday interuptions.
Instead, offer to be the maintenance number. Offer to put lockboxes on doors for showings. It is usually everything involving travel owners dislike. Most can handle the phone calls.
Property Management dine tight is super hard. You wouldn't believe the number of processes, softeare, and checklists we use. There are so many variables and crafting responses to them all takes time and experience.
Best of luck in whatever you decide.