Austrian here, coming here is a good idea :D
If it's about continuing from DIY then you'd need to get it prescribed, but getting to HRT through official means really isn't too big of a deal, especially if you don't have to jump through hoops like "too depressed, other undiagnosed and untreated mental health issues, former wrong diagnoses, ..." that I had to go through.
I was at Courage Vienna (courage-beratung.at) for psychological counseling and they helped so much with everything.
For HRT you'll need a few, what we call "stellungsnahmen". Basically just from different people a paper saying " yes, you're trans and yes, you should get HTR". You'll need that from
- a psychologist (with Transgender diagnosis)
- a therapist (courage can do that)
- a psychiatrist (usually with another diagnosis)
With all of that, you can get an appointment at an endocrinologist and get HRT.
Keep in mind tho, while there is free healthcare in Austria, there are also private doctors that you have to pay, but in return you get way shorter waiting times. You can also send the bill to your health insurance and get some of the cost back.
If it's about continuing from an existing prescription then it should be as easy as sending your papers to your health insurance, but i'm not too sure how exactly what they need and if you need to do more...