If you restrict Druids to non-metal armor as listed in the Player’s Handbook, they can get locked out of the strongest medium and heavy armors:

If you run this rule as written (RAW), the only medium armor you’re proficient with (Hide) is worse than your best light armor option (Studded Leather), due to the Dexterity cap. Depending on your Dexterity investment, this may not make a difference. However, uninvested Druids would certainly like to make use of the entire gamut of medium armor, which offers up to an additional 3 AC (or 2 AC without imposing disadvantage on Stealth checks).

This restriction is thematic. There’s no balance concerns from giving Druids access to all armors. D&D Lead Rules Designer Jeremy Crawford explains:

What happens if a druid wears metal armor? The druid explodes. Well, not actually. Druids have a taboo against wearing metal armor and wielding a metal shield. The taboo has been part of the class’s story since the class first appeared in Eldritch Wizardry (1976) and the original Player’s Handbook (1978). The idea is that druids prefer to be protected by animal skins, wood, and other natural materials that aren’t the worked metal that is associated with civilization. Druids don’t lack the ability to wear metal armor. They choose not to wear it. This choice is part of their identity as a mystical order. Think of it in these terms: a vegetarian can eat meat, but the vegetarian chooses not to. A druid typically wears leather, studded leather, or hide armor, and if a druid comes across scale mail made of a material other than metal, the druid might wear it. If you feel strongly about your druid breaking the taboo and donning metal, talk to your DM. Each class has story elements mixed with its game features; the two types of design go hand-in-hand in D&D, and the story parts are stronger in some classes than in others. Druids and paladins have an especially strong dose of story in their design. If you want to depart from your class’s story, your DM has the final say on how far you can go and still be considered a member of the class. As long as you abide by your character’s proficiencies, you’re not going to break anything in the game system, but you might undermine the story and the world being created in your campaign. – Jeremy Crawford, Rules Answers: March 2016

Balance is based on armor proficiency. The fact that Druids don’t have access to certain armors within a tier is a vestigial rule. There’s nothing that requires armor of a particular AC to be crafted from metal.

Natural Metals?

Metal may not break the Druid’s natural armor restriction. It might just be that your Druid has no aversion to metal, just non-natural sources.

Consider the existence of (non-construct) creatures which have a metal hide, such as the Gorgon. You can also homebrew certain enemies to have metal hides. The Bulette is a natural.

Quests to gather these types of reagents can be more rewarding since they are player-driven. They also build relationships between characters who will be quick to remind the Druid player that they owe them one for the robot rodeo.

Non-Metal Armors

Assuming you want to keep the non-metal restriction, let’s take a look at the different armor types and see if they allow for non-metal, or if we can build a homebrew around them. Armors which are metal by RAW will be listed in red.

Light Armor

There’s not much concern that the druid is going to be prevented from using light armor.

Padded. This armor is made from cloth, which is derived from plants.

Leather. This armor is made from leather, which is derived from creatures.

Studded Leather. While studded leather’s rivets or spikes can be made from metal, they can just as easily be made from stone or bone.

Medium Armor

Medium armor is where you really start to see differences in what Druids can access versus other classes with medium armor proficiency. This matters because there’s a wide range between potential AC classes and differences in whether disadvantage is imposed on Dexterity (Stealth) checks.

Hide. Like leather armor, but thick and furry.

Chain Shirt. This is the first armor which indicates that it’s made of metal. It’s worn with plant and animal-based clothing to dampen the sound, but the essence of the protective effect is that it’s metal. Technically, leaving this out isn’t a game changer, since the chain shirt occupies a middle tier among medium armor. Breastplate offers a higher AC bonus, with no disadvantage penalty or strength requirement. Since this represents a malleable armor, I like representing this armor with a web of gnarled woody vines. The Druid must undergo a ritual where they meditate in a grove while the armor grows over them. The ceremony warrants some reagent collection to seed the growth of the magical wood. Perhaps they need to collect a sample of (each kind of) blight (Twig, Needle, Vine),

Scale Mail. As we hard from Jeremy Crawford, feel free to take “scale” very literally. Any time the party slays a dragon, at least one player is going to try and harvest part of it. Let them use those scales to create some Druid-friendly armor. I like to season this with a little resistance of the slain dragon’s breath weapon type.

Breastplate. While the breastplate description does describe a metal plate, it’s easy to think of something else this could be. Even something as rudimentary as a sandwich sign can fill this thematic gap, depending on the humor level of your campaign.

Half Plate. This armor’s description also describes metal plate. It’s going to require some homebrewing, as half plate occupies a unique niche. It’s the only armor with 15 + Dexterity modifier AC, and it imposes disadvantage. For druids, it makes sense to use chitin as a different type of hard shell. I like to make the character chase down a reagent for this type of crafting. This gives a real reason other than monetary cost why such an item is not available until a certain tier. Consider using an Ankheg (CR 2), Chuul (CR 4), Umber Hulk (CR 5), or Drider (CR 6). There’s some other chitinous creatures, but they’re a little tougher so those reagents may be better invested on stronger armor.

Heavy Armor

While Druids don’t get heavy armor proficiency, you might take a multiclass or a feat (Heavy Armor Training) to gain access to heavy armor. These are all customarily metal armors, which means we’re going to need to do some homebrew for heavy armor druids.

Ring Mail. This armor’s description does not actually prescribe metal rings. Rings can still be made from other materials, such as hardwood, stone, or bone. Besides flavor, there’s not much homebrewing required here.

Chain Mail. Unlike ring mail, this armor’s description specifies that the interlocking rings are metal. It’s a bit strange to imagine chain mail being composed of any other material, though technically possible. Wood can be carved into interlocking rings by a skilled artisan, though small enough ringlets to simulate chain mail would require an exceptionally skilled craftsman. For harder media like stone, this seems even more ridiculous. We’re going to need to patch something in here, since an AC bump and a cheaper price is not the only thing keeping this from being splint armor. It also has a lower strength requirement (13 STR) than splint (15 STR). Like the chain shirt, we can represent this with gnarled vines. However, it might require a reagent from a tougher creature, such as a Shambling Mount (CR 5) or Treant (CR 9).

Splint. Splint armor is made from strips of metal backed with leather. This is probably the easiest homebrew, as you can swap the metal plates for superhard wood. In earlier editions, Druids had access to a transmutation spell that would allow them to craft ironwood.

Plate. Plate armor is composed of interlocking metal plates. Like half plate, here’s no reason these plates need to be metal. However, they might require a tougher chitin. Chasing down tougher reagents gives a real reason other than monetary cost why such an item is not available until a certain tier. Consider using an Remorhaz (CR 11) or Purple Worm (CR 15), although you can open this up to some of the half plate chitins if you want to give players access earlier in the campaign.