For the public health clinic, recommend you contact Nurse Natalie at 501-226-3668 or 501-226-2555. The town clinic is now located near the high school.
For the Lions Club clinic, recommend you contact Severo Guerrero, phone 501-226-2324, E-mail: firstname.lastname@example.org. Severo is past president of the Lions Club and current San Pedro deputy mayor. He could direct you to the LC clinic contact.
You can call the island on a Sam's Club AT&T calling card for just under 25 cents US per minute; the card shows 3.47 cents per minute when used for US calls.
I'm not versed in the work permit issue for volunteers, but I met students from at least 2 US colleges/universities the past 2 summers that were conducting projects on the island. Both groups worked with the Town Board and/or Hol Chan in some capacity, so probably someone of some authority blessed their presence, but I don't know if it was through work permits or not. Mito at Green Reef may know. Good luck and thanks!