Is it vital for Jews to attend synagogue every Sabbath?
I have found no source that dictates that Sabbath services are mandatory. Until the early 20th Century there was no separate Sabbath service, other than some variations on the daily service. Because the synagogues sprung up during the Babylonian captivity, after the destruction of the First Temple, many practices mirrored that of the Temple service. This included daily prayers. While the prayers on Sabbath were slightly different in some cases, Jews were still expected to assemble every day, not just on Sabbath.
In more modern times some Jews have emphasized Sabbath assembly, although traditionally the Sabbath was set aside for families at home. There may be several reasons for emphasizing this one day over the others in synagogue. In modern America, many men (and in Orthodox synagogues only the men make up a "minyan", a group of ten required before certain prayers are said) can not attend the morning or evening daily prayers because of work requirements. Thus the practice of ensuring they are present on Sabbath. Likewise, that is the day that children who don't attend Synagogue School may be more likely to be available for Torah study. A third, less likely, reason might be the influence of Christian worship on Sunday, so that some Jews wanted their own assembly just as some now include a Chanukkah bush when their neighbors put up Christmas trees.
By the Roman times there were those who habitually attended Synagogue on Sabbath. It was Jesus' custom to do so (Luke 4:16). It was also Paul's (Acts 18:4). But this does not imply that it was not their custom on other days as well. It is true that Paul specifically talks about the Prophets being read "on the Sabbath," (Acts 13:27)but that may be because the Sabbath was the day that the weekly readings were changed.
Sunday, however, is not "the Christian Sabbath." Christians don't generally keep Sunday like Jews keep Sabbath. So, while Christians may be required or encouraged to assemble on Sunday, such has never been the specific practice of most Jews in relation to Sabbath.