The best information we have as of today is that Grants.gov will continue to operate normally during a federal government shut-down. The site is hosted and staffed by a government contractor at a third party location. These operations are fully funded, so all indications point to grant applications being processed normally by the Grants.gov servers.
Most of the main sponsoring agencies, like the NIH, have systems which automatically retrieve grant proposals from Grants.gov. As long as Grants.gov is accepting applications and as long as the automated agency systems are retrieving them, proposals will be submitted normally.
However, anything that requires human interaction would be subject to delay. Technical issues with a submission, for example, that would normally route through a help desk escalation process would not be likely to receive attention during a shut-down, owing to staff and contractor furloughs. This good news for Cayuse sites, since errors or problems on proposals are quite rare.
The official guidelines from the White House to the federal agencies are posted here (PDF). The only services and web sites that would continue operating are "excepted activities," meaning those that deal with public safety.
This article from Federal Computing Weekly offers reporting on questions surrounding other types of government web sites.