How many times per year are allergy shots given?

Once a month. Allergy shots are given once a month if you are on maintenance treatment. To get there you have to start with shots once a week for about nine months or so and then once every two weeks, three weeks, and finally once a month. There are "rush" protocols to reach maintenance which are faster, but also more likely to cause side effects, like anaphylaxis(serious allergic reaction).
Weekly to monthly. Allergy shots, or "allergy immunotherapy", start out a low concentration and are built up to a target, or maintennce, concentration and dose. Schedules for building up to that maintenance dose vary according to your provider's comfort and expertise. There are rapid build-up protocols available to cut down the time needed to reach maintenance doses.