So I tried to take on a small group of bandits, and during the fight I had my right eye cut out. The DM told me it can't grow back, so I asked him about anything I could replace it with. He said maybe some sort of magical eye would work. Are there any official magic eyes?
Your best bet is the Ersatz Eye, a common magic item listed on page 137 of Xanathar's Guide to everything:
Wondrous item, common (requires attunement)
This artificial eye replaces a real one that was lost or removed. While the ersatz eye is embedded in your eye socket, it can’t be removed by anyone other than you, and you can see through the tiny orb as though it were a normal eye.
Magic items that let you see
The classic official eye replacement is the Eye of Vecna, though it's an artifact, and a nasty one at that (DMG 224):
To attune to the eye, you must gouge out your own eye and press the artifact into the empty socket. The eye grafts itself to your head and remains there until you die. Once in place, the eye transforms into a golden eye with a slit for a pupil, much like that of a cat. If the eye is ever removed, you die.
Properties of the Eye. Your alignment changes to neutral evil, and you gain the following benefits:...
The Eye of Vecna also grants you a number of powerful abilities, too. It seems unlikely that you could access such a powerful item if you're getting seriously hurt by bandits, though.
Another less rare item that helps you see is the Robe of Eyes, which is only Rare (DMG 193):
This robe is adorned with eyelike patterns. While you wear the robe, you gain the following benefits: The robe lets you see in all directions, and you have advantage on Wisdom (Perception) checks that rely on sight...
In addition to official items, I have created other replacement eyes for my players. I realize that's a little outside of the scope of your question, but I suspect that such eyes are a common kind of homebrew.
A spellcaster can regrow your eye
Instead of a magic item, DMG 272 details variant rules for losing and restoring an eye:
Lose an Eye. You have disadvantage on Wisdom (Perception) checks that rely on sight and on ranged attack rolls. Magic such as the regenerate spell can restore the lost eye. If you have no eyes left after sustaining this injury, you're blinded.
The DMG presents these rules only as options, so they aren't binding and they require your DM's approval. Still finding someone who can cast a 7th level spell for you will certainly be easier than tracking down the Eye of Vecna.
The regenerate spell is described as follows:
You touch a creature and stimulate its natural healing ability. The target regains 4d8 + 15 hit points. For the duration of the spell, the target regains 1 hit point at the start of each of its turns (10 hit points each minute).
The target's severed body members (fingers, legs, tails, and so on), if any, are restored after 2 minutes. If you have the severed part and hold it to the stump, the spell instantaneously causes the limb to knit to the stump.