Should the United States allow Syrian refugees in to the nation without fully vetting them first?

Ahhh... NO.
Nor should the United States allow anybody from anywhere in without vetting them. It isn't just Syrians that cause trouble. So far in the US, no one who has committed a domestic terrorism attack was Syrian, and ISIS isn't just in that country. There are ISIS affiliates in Algeria, Nigeria, Libya, Egypt, Somalia, Gaza, Iraq, Lebanon, Afghanistan, Pakistan, Indonesia, and the Philippines, and I am sure I missed a few. Not to mention sympathizers from all over. And ISIS is not the only threat.

Shouldn't allow them in the first place. But, if the powers that be do want them to show up, then they need to be vetted BEFORE they leave their own country.

We should not allow them entry under ANY circumstances!