Some do. I would try: general practitioners, endocrinologists, OB/GYNs, and Planned Parenthood locations. Beyond that, you will probably not have any luck. You may very well end up needing to get a referral, or to save money until you can afford both the trip and hormones to go somewhere that offers HRT through informed consent.
If you are near any universities or colleges (not community colleges, something 4 year), you could check there as well. If they have an affiliated hospital, they might offer it with a referral. They may have LGBTQ student groups that you could ask. Every campus also has some kind of student health center and you could call and ask who they refer students to if they come in requesting HRT.
Also, I'm not sure how you're asking the folks you're calling, but what I did when I started looking was ask if they offer HRT for transgender patients, and if they did, what they require in terms of referrals in order to start treatment.
And on the topic of those youth support groups, you could always get the contact information for whoever is in charge of the groups and ask if they know anything about local resources. It's not just a bunch of kids hanging out, somebody's there watching them and directing their activities.