How much soil for a 4x8 raised bed?

A "4x8 raised bed" is a footprint, not a volume. The height is the number the phrase leaves out, and it is the one that sets your order.

Greenes Fence sells that same 4 ft × 8 ft bed in three heights and prints the capacity on each: "Holds up to 28 cubic feet of soil" at 10.5 in, "Holds up to 37.3 cubic feet" at 14 in, and "Holds up to 46.7 cubic feet" at 17.5 in. Gardener's Supply's 4 × 8 Deep Root bed, with 14.5 in of planting depth, "holds 1028 quarts (40 cubic feet) of potting mix". Same name, same footprint, 28 to 46.7 cubic feet — a factor of 1.7.

Bought loose, that is the gap between a one-yard order and a two-yard one: 28 cubic feet is 1.04 cubic yards and 46.7 is 1.73. So measure the inside height of your boards and put that in the depth field, rather than trusting the number in the product name.

The second missing number is printed on the bag. Divide Gardener's own pair — 1028 quarts against 40 cubic feet — and you get 25.7 quarts per cubic foot. That is the dry quart, 67.2 cubic inches. Espoma's potting mix is stocked in "one cubic foot (25.71-quart) bags", which is the same ratio from a second maker, so it is the measure bagged mix is actually sold by. The liquid quart that most conversions reach for is 29.9 to the cubic foot, and reaching for it makes you buy 16% more than the bed holds. It is also why the bag list above, which is in cubic feet, will not match a bag labelled in quarts — divide the quarts by 25.7 first.

At this bed's 12 in, 32 cubic feet is 823 dry quarts, or 26 of the common 32-quart bags. The liquid conversion says 30. Four bags is most of a wasted second trip.