Weber is found very widely throughout the German-speaking parts of Switzerland.However, the family had long-standing citizenship in only a few towns in Graubunden.
A good place to start when one is researching Swiss surnames is always the Register of Swiss Surnames, which has been available free online for a number of years at http://hls-dhs-dss.ch/famn/?lg=ehttp://hls-dhs-dss.ch/famn/?lg=e
It shows where in Switzerland there is a family with a given surname and the age range when the family obtained citizenship.This lets you narrow your search, but of course does not mean that your ancestor was in one of those particular towns :-)It also does not show locations where the family previously had been present but no longer has any members with citizenship.Also remember that someone could have citizenship in one town while living in another town or even out of the country.Vital records would generally be transmitted back to the church of the town where one had citizenship, but there usually would be a delay, which at times could be many years (say until a child married or some other significant event)!
The results for Weber in Graubünden are
Surname Canton Community Citizenship obtained Place of origin Weber GR Arvigo 1920 (D) Weber GR Augio 1942 (D) Weber GR Breil/Brigels b Weber GR Casti-Wergenstein 1893 (D) Weber GR Cazis 1933 (D) Weber GR Chur 1852 (Langwies GR) Weber GR Davos 1918 (D) Weber GR Grüsch 1861 (F) Weber GR Langwies a Weber GR Mathon 1821 (A) Weber GR Saas a Weber GR Sils im Domleschg 1899 (D) Weber GR Untervaz b Weber GR Zizers 1825 * Legend: a = before 1800; b = in 19th century; c = in 20th century.