Ooo! A question about my industry that I actually have a competent answer for!
For Avoidant PD or even related disorders you might want to start with a psychologist or both a psychologist and a psychiatrist.
Get a list of covered psychiatrists/psychologists from your insurance company (probably can get this from the insurance's website too). Then call and ask them if they are sensitive to polyamorous or non-monogamous issues. If they have no idea what you are talking about, try the next name on the list. You also might get a receptionist, so make sure you speak to the actual person your appointment will be with. You can ask other questions over the phone too. And if you make it to your appointment and things don't work out with one professional, you can certainly switch.
There usually is a wait time to get in to see mental health professionals so if you do end up switching keep your appointments with the not so good match until you can get in to see another professional. It is perfectly ok to make sure you are seeing the professional that is going to be the greatest help to you.