I normally buy frozen bags of vegetables from the supermarket, but I currently don't have a freezer to store them in and have been putting them straight in the fridge instead.
How long will they keep there and still be safe to eat?
The cell walls are now damaged from the freezing, so they cannot keep like whole, live vegetables. They are in the same position as if you had damaged the cell walls by heat = cooking them. So, apply the rule for cooked food.
They will be safe for 3-5 days.