Vancouver Island and the City of Vancouver or entirely seperate entities.
The City of Vancouver is located on the British Columbia mainland in the south-west corner of the proivince.
Vancouver Island is a large island, (286 miles long, 30-50 miles wide), located 30 miles off the mainland of British Columbia. The provincial capital city, Victoria, is located on Vancouver Island. Access to the City of Vancouver is usually by ferry (1.5 hours), or by by aircraft.
There are no restrictions on foreign citizens buying real estate, although the Vancouver real estate market is very pricey.
As a US citizen you could enter Canada for up to 6 months at a time as a tourist. You would have to satisfy Immigration that you have intentions of leaving the country. As a tourist, you will not be allowed to work.
If your plan is to move to Canada on a permanent basis with the option to gain employment, you'll have to explore status as either a Permanent Resident, or holder of a work visa.
Generally, those applying for Permanent Resident status do so from outside canada, although there are circumstances under which you can apply from within canada.
You can find all the information at the Government of Canada Immigration web site:
www.cic.gc.ca