Does trampolining burn as many calories as running?

Might do it. With running, or walking, seems to be a calorie expenditure of about 150 calories per mile in some adults. Do not know that we have an hourly calculation for exercise on trampoline, but vigourous exertion could use 700-800 calories per hour. Any aerobic exercise is going to burn calories.
Jumping. It depends on how vigorously you jump and how fast you run. If you jump hard for 1/2 and hour and run at a 17minutes/mile pace then jumping will burn more. If you run at a 7 minute pace and leisurely jump then running will burn more. Likely 100 calories per half hour for basic jumping. About 100 calories per mile for jogging.
Exercise. There is no right answer to this. There are too any variables. But any exercise is good!