Mrs. Claus opens "The Year Without a Santa Claus" by claiming the eponymous year took place "before you were born". Seeing as the movie was released in 1974, this means the year must have been before then.
Bounding this on the lower end is the presence of ice hockey - mentioned by Heat Miser - and the use of telephones. Ice hockey was invented in 1875, while Alexander Graham Bell built the telephone in 1876, meaning the year must post-date these. These figures give a range of approximately 100 years during which Santa may have taken his holiday.
Yet, narrowing this further is the presence of a December calendar counting the 1st to a Wednesday. Between 1876 and 1974, only the Decembers of 1880, 1886, 1897, 1909, 1915, 1920, 1926, 1937, 1943, 1948, 1954, 1965, and 1971 started on a Wednesday.
But still this can be narrowed further.
When Santa set out that Christmas Eve, we see what appears to be an almost full Moon in the sky. Within the years listed, only 1920 had a full Moon on Christmas.
Ergo, 1920 was the year without a Santa Claus.
