Instead of buying bags of potting soil, it is usually cheaper to buy the ingredients and blend a custom mix.
I can get 3.8 cu ft of peat moss, 4 cuft of perlite or vermiculite. These are expensive, but cheaper than the equivalent you will need in bags of potting soil. Some people use coir instead of peat. It holds more water but it is more expensive.
You can probably get it from the agricultural wholesaler. Most of the ones here also sell retail and you can get a discount if you set up an account. If you are not there yet and have gardening friends, maybe they know where to get the best buys. If they have an account maybe they can get it for you.
Compost from the city green waste recycling facility. Cheaper by the scoop than by the bag. You might even be able to get mulch free, either from the compost facility or by calling tree trimmers in the area.
Horse manure is usually free. Most stables will be glad to have you take it away, you just have to ask. The down side is if it is not hot composted first, it can have a lot of weed seeds.
Since your weather is brutal, it is probably a good idea that you are making a deep bed. I'd go a little deeper my self, I like 18 inch depths because some plant roots do go that far.
Around here, everything is expensive, so anything I can get free I will use. Lumber would be especially expensive.
Use rocks to form raised beds. Here hollow tile is cheaper than lumber and lasts longer. Dry stack the tile and stabilize the tile with rebar and fill the holes in the tile with dirt, stones or concrete. Other choices are recycled wood from remodeling projects, railroad ties, and pallets (most companies throw them away so they usually will give them away or be low cost.) They will last a few years and have to be rebuilt, but by then you can probably find a more permanent solution.
I would also leave the bottom of the box open. Break up the soil with a fork, lay down the cardboard or newpaper and then fill the box.
It would have been better to start the bed in the fall, but if you can wait to plant you could save a lot of money by sheet composting instead.
Layer with the cardboard newspaper. Kitchen wastes, coffee grinds from starbucks ( just ask and they will save it for you. A lot of people ask so you may have to stand in line for it.) Toss in leaves, start a compost pile
get the free tree trimmings. You can have a usable bed in about 6 months. You can add manures to this pile because it will take 6 months to be ready which is just about right for it to be safe even cold composted.
While you are building the bed, you might want to oufit it so that you can add hoops on it. You can cover it with shadecloth during the summer and frost protection later.
https://extension.oregonstate.edu/lane/s ... osting.pdf
You could also go the permaculture route and build a keyhole garden. The garden is built using any material that is readily available. It is a drought tolerant design. The central basket can be made out of wire instead, it will last longer than a stick basket. Composting made simple and water efficient. You can even add a hoop structure for shading.