You might want to read through this thread. http://www.englishforum.ch/insurance/210112-swisscare-no-longer-valid-phd-postdoc-share-us-what-you-know.html
Swiss health insurance will cover you immediately, in fact it's usually backdated to the date of your arrival in Switzerland so any medical bills you incurred before you got the policy should be able to be claimed back.
Why enquire about cover in St. Gallen when you're based in Vaud?
Swiss health insurance cover is recognised by all hospitals, but depending on the type of policy you take out you may be restricted to certain doctors/hospitals.
Note that, from what I can see on Swisscare's website, you have to pay up first yourself and then claim the costs back from them. This may not be ideal for you.