Yes, folks can refuse to work with you. It's strongly suggested to seek out providers who are knowledgeable about transgender healthcare so that you can avoid a lot of headaches. If your friend has had a good experience with the therapist that runs the group you're going to, I'd definitely look into seeing her. If you like her demeanor at the group, maybe try talking to her afterwards and see if she is taking new patients.
A therapist can refuse to refer you or put stipulations on giving you a referral. The most important thing for them is making sure that what you are asking for is something that will actually alleviate the feelings that you have, and that you are in a good place to do it. They have to rule out that it isn't another condition causing you to have dysphoria-like feelings. The second part, about being in a good place, seems to be hit or miss as to whether or not they care about it and to what degree. Most would like to see that you have some kind of support system in place, that you have a plan in mind for your transition and have possibly been taking steps towards transitioning socially, and that in general your affairs are in order so that you are better able to deal with the stress of a transition. But some don't care about any of this, or only care about parts of it. If you're in kind of a tenuous place, they will probably give you a referral but encourage you to stay in therapy with them on a regular basis.
A doctor can refuse to prescribe you hormones. Some will tell you they don't offer it as a service. Some will be honest and say that they don't know enough about it to feel comfortable doing it. If they've prescribed to trans patients in the past, they will likely prescribe to you as long as you are otherwise healthy enough for it. If you have pre-existing conditions that could worsen on hormones, they may be hesitant to prescribe if you are not or will not manage that condition.
Once you get to the point of surgery, it's rarer to find people saying no. Especially if you're using the same therapist throughout your entire transition, they'll have a good idea of what you want and whether or not this is a good move at a good time for you. On the doctor/surgeon side of things, more common is "Yes, but" - as in yes, they'll perform the procedure on you but you need to do X, Y, and Z before that can happen. They have a whole laundry list of criteria they need to meet when they're submitting our cases to insurance, and they want to make sure they cover all their bases so that they can get paid and you can get what you need at minimal cost. They also want to make sure that they're putting you in the best position to have a low complication rate and to be happy with your results.
If you can get connected to a good therapist and you can find a local HRT provider (or are willing/able to travel to a larger city), you will likely not have any difficulties when it comes to surgical referrals. It's just those initial two connections that people sometimes have issues with.
