All good advice so far. IMO the best way is to do all of the above, and just keep it in the back of your mind. I've gotten mulberry from a park once (was already cut and stacked by a dumpster), driving down the road after the highway dept trimmed and found chokecherry and buckthorn, asking friends, colleagues, neighbors, landowners etc.
Remember even if you cut today, you won't be shooting for a while, so if you've got a few bucks I suggest finding a decent lumber yard.
Yeah you can go to the big box stores, but there wood is a lot more expensive, lower quality, and at least in my area seems to be dry as a bone. A decent lumber yard will have a much better variety, in a variety of dimensions, and better quality, for less money.
The blue store has red oak at near $8/Bdft, that's too much IMO.
In contrast, I bought some perfectly straight 8/4 ash from an indoor lumber yard near me for like $5/bdft. And that was s2s not rough cut.
Also, I know in this area at least, some of the tree services also saw, air dry, and sell some common domestics like oak, maple, pine and tend to also be very reasonably priced.
If you can resaw the lumber, you can make a number of bows out of a 6" wide 6' long piece of 8/4 ash and I think that would run like $30 around here.
What I did years ago is just buy a couple 10' Ipe 2x4s and maybe 10bdft of hickory .... probably spent a bit over $100 but I've built so many bows from that stash I've lost count.
My average tri-lam is built using a total of ~1bdft of material plus the handl/riser material. And that includes the waste from the table saw kerf. So I doubt they run more than maybe $10 in materials unless I get fancy.
Just one more example, 6" wide 6' long 8/4 hickory would yield 15-16 1/4" thick backing strips and could probably be bought for $25-30.
The nice thing about board bows is you can finish one in a pretty short time with only the most basic tools. Again, if you can resaw a backinstrip and a belly lam and add a powerlam in between, you can really minimize the amount of wood you need to remove from the belly and it can go pretty fast.
Sorry for the essay.
Post script edit: MacBeath Hardwoods in SLC looks like a reasonable place to start.