Or can you do just fine without it?

personally i cannot have one without the other.

Stealth makes archery gameplay too boring and too easy IMO.

For non sneaking warriors in armor I use a crossbow.

Crossbows aren't good for stealth as they are noisy and give away your position but are good for giving a warrior a ranged attack.

My very first character in the game, and the one I have the most hours logged in, wears Light Armor, primarily uses a bow, and never sneaks. It demands a lot of good positioning and forcing your enemies to stay away, and it also takes a lot of practice with aiming, but it's very rewarding. It also snowballs, getting stronger and stronger as you invest perks. It gets to the point that, if you manage it correctly, you can handle several Draugr Deathlords in melee range with just your bow. Without getting hit.

That said, you do need to be a good shot, know how to utilize terrain, how to pick targets, and when to pull out a melee weapon if things get extremely hairy. It's also sometimes a good idea to pack up and bolt for another good spot to shoot your targets from. It's also helpful to know all the different benefits of your bow. For example, the Hunting Bow doesn't hit hard, but it draws faster than any other bow in the game. Typically, bows made of Light materials draw faster with less damage than bows made of Heavy materials, though they still differ from bow to bow.

If you do manage to get the intricate parts of combat archery down, and invest a healthy amount of perks into it, you can be truly devastating.

My current character is an archer who uses an axe when she runs out of arrows. She does not sneak at all.

I am playing an archery character for the first time, and I must say, I find sneaking to be pretty useless. You really get most of your mileage from sneaking when you are in caves, but when you are perked out, it barely matters from a combat perspective. I like to use my bow on targets that like to stay at range, especially mages. I use the bound bow as well, so having to summon it kind of destroys your sneaking options.

I guess to sum it up: I don't think they go hand in hand, but they do help each other out.

Go together like drum n' bass Sir.

When using a ranged weapon you need to be as far away from the enemy as possible and not subject to their fire.

And it takes time to drop your bow and draw your melee weapon.

Also good for 'WIlliam Telling' merchants during a slow afternoon at the market.

I sneak often. But never put any perks into it.

There is a crit modifier if you hit with the eye fully closed.