It means the snow can last many days in a row, many days at one time. At a time, is when something happens in a short space of time; it can be several minutes, days, weeks months but all one after the other rather than irregularly. He can play several games at a time. It rained for several days at a time. I study for 3 weeks at a time. How many hours sleep do you have at a time?

First of all, know what do the words mean. "Blizzards"- Heavy snow storms; "May"- Here, used for uncertain possibility; "For Days"- Here, used for saying how long something lasts; "At a time"- During any one period. Now, your sentence means "Heavy snow storms [Blizzards] may last for days [- Talking About How Long The Blizzards Last] at a time [During Any One Period]." If you still didn't understand, let me know.

At a time : if someone deals with things one, three, ten etc at a time, they deal with them separately or in groups of three, ten etc ( Dictionary Longman)