I used to get all my groceries by bicycle until recently. Did it for years. I used two sturdy bicycle panniers on a bike rack. I was always amazed how much I could pack in - so were some of the store employees. I could easily get in 3 bags worth. Depending which store I used, I'm 3- 5 miles away. I did it year round.
In Portland it's very doable because the main roads get cleared up fairly quickly after a storm. Even after a bad one they will be clear usually after a day, 2 at the most. Side roads were a different story. I never ride until they are clear [I still ride year round].
I only used a simple Trek Electra bicycle without any motor assist. Before that I used an LL Bean Runaround Cruiser. The Electra is a beefy road bike, the Cruiser is a hybrid. I don't even use stud tires [in winter]. You don't need them if you stick to the main roads.
I'm very curious what road conditions are like for Alice since that is more inland, more snow, and probably less plowing. I have heard you can easily ride safely on packed snow with stud tires but I've never tried it.
Denise