When fly fishing for stripers or just about anything else, you tend to carry lots of small and not so small things. Objects like flies, fly boxes, tippet, bug repellent, an extra reel or spool, maybe a light jacket, water bottle, polarized glasses, hook hone/sharpener, etc. Most fly fishermen use a vest to help carry and organize these items.
I even use a vest when I'm in a boat. I like things right where I can reach them. A tackle box isn't as convenient, plus fly rodders are often in the water, while the tackle box is on shore. Flyfishermen often cover a lot of territory too and wearing your gear is much more convenient than carrying it in a box that you need to pick up and put down periodically as you move around and fish, whether from shore, while wading, or even in a boat.
A few fishermen don't like fly fishing vests and prefer a fanny pack, chest pack, or backpack. Whatever works for you is fine, although I find a backpack inconvenient as you usually need to take it off to access it and a fanny or similar pack to be too small. I will admit that when wading for bonefish I usually don't use a vest, just a few accessories I toss in my oversized fishing shirt top pockets.
Some kayaks prefer a chestpack, perhaps combined with an inflatable PFD. This combo is good for other fishing besides just kayak fishing as well, for example off of rocks etc. Cabelas makes decent and modestly priced chest packs and inflatable PFDs. One reader also wears a rain jacket with big pockets to hold other items as there are no PFD or flotation vests that can hold big fly boxes (also the Finsport Streamer Wallets are very good for storing striper flies).
While any vest can do, I have strong preferences for three features:
Big Front Pockets - I like a couple of oversized front pockets, which you normally don't find on vests designed for trout. Why big pockets? To fit bigger fly boxes that have bigger flies than most freshwater fishermen use.
Shorty Style - I wade a lot, including in the surf, and prefer a shorter style vest so that it's not continually getting wet on the bottom. Yes, my vest gets wet often and I don't worry about it, but I don't like it dragging in the water and soaking the bottom pockets continually.
Roomy back pocket - In the back pocket, the one you need to remove the vest to access, I carry an extra reel or spool, and often a pullover or polypropylene top as well as a small water bottle. I like as big a back pocket as possible, and most vests seem to have one.
Vests last a long time so it's worth investing in a one you like. Mine, a brand no longer made, has been with me over a decade and I bought for about $25 at a clearance. You can spend a couple hundred dollars on a great Simms Vest or even more, or much much less on an inexpensive vest that gets the job done.
My preference is to try vests on in a local shop before buying, if that's possible, and see how well large fly boxes fit into the pockets. I also like a vest large enough so I can wear a couple of layers of warm clothing either under or over the vest, which means I like big vests. Remember you'll be casting while wearing your vest, so you want lots of room for movement.