Hi Chris,
Schwägalp
This site is around the valley station of the cablecar to the Säntis, the highest mountain in Appenzell. The forest behind the big restaurant holds all alpine forest species like Willow Tit, Coal Tit, Crested Tit, Three-toed Woodpecker (rare), Black Woodpecker, Green Woodpecker, Dunnock, Crossbill, Hawfinch, Bullfinch, Tawny Owl, Pgymy Owl (rare and difficult in May), Common Treecreeper, Hazel Grouse (difficult), Woodcock, Ring Ouzel, Nutcracker and Citril Finch.
The meadows around the restaurant hold Rock Pipit, Whinchat and Northern Wheatear.
Säntis
The mountain itself can be quite crowed. Alpine Chough is always present, Alpine Accentor and White-winged Snowfinch are both possible. Wallcreeper is present but hard to find.
Check on the internet:
http://www.saentisbahn.ch/pages/d/index.php
Ebenalp-Wildkirchli
This is an excellent site for Wallcreeper. Take the cablecar to Ebenalp and walk to the famous cave of Wildkirchli (= wild church). Scan the rocks here for this wonderful species. It's worth to come early, before the crowd.
http://www.ebenalp.ch/sommer/sehenswuerdigkeiten/
Rheindelta, Austria
Not in Appenzell but very near is the Rhinedelta. Pass the boarder in St. Margrethen and drive on in direction Bregenz. Just before you pass over the Rhine (the big one, not the old Rhine at St. Margrethen) go left until you reach a parking lot. Walk out from here (about 3km) into the Delta. Waders, Great Reed Warbler, Terns, Ducks, you can find anything here. May is a very good time for vagrants.
Toggenburg
A region just south of the Appenzell with Wildhaus as central village. The forest here hold many good species like Hazel Grouse, Pygmy Owl, Tengmalm's Owl, Black Grouse and even Capercaillie. If you plan to go here, PM me for exact locations.
I hope this helps. Greetings from Switzerland
André