As rgatijnet1 said, the brakes will work fine aside from the antilock system not kicking in, the same as if the vehicle simply never had antilock brakes. I personally would fell perfectly fine finishing a trip under those circumstances and getting whatever the problem is sorted out once home, especially if it were practical to avoid driving in poor weather or slippery conditions. It is not very hard to avoid snow and ice this time of year.
(A bad or misaligned or poorly-connected sensor is a likely culprit, but there are a few other things that it possibly could be as well, including something with the control module, a failure in the BS valve and pump system, etc.)