You are over watering. Cut back on the watering and give them indoor (not outdoor) sun until they recover. Then slowly transition them to outdoor sun starting with shade.

They definitely need indoor/indirect light when this young. I would also suggest spritzing them with water several times a day. As Nik said, they have tiny little shallow root systems, so they can only absorb water from the upper most layer of soil. Saturatering the soil doesn’t do much good at this point

If they do not perk up after being out of direct sunlight, then it is overwatering.

Is the mix able to re-wet easily? There are some soils that look moist on the surface but stay dry below. Best way to know if it's under or overwatered is to stick in a wooden skewer or like the Popsicle stick. Leave it for a few minutes and check if it absorbed any moisture or if a lot of damp soil pulls up with the skewer. The seedlings should also be in shade/indirect sunlight at all times when it's that small too. I also think seedlings are easier to care for if you grow them in long and narrow containers. It can grow its roots longer and will less likely be overwatered since its narrow.

I'm growing some citrus from seed too. They can die from sunlight even if they're properly watered since their root system is too small to keep up with its water needs as nik said. My seedlings are outdoors and even smaller than yours but I have them inside a mini greenhouse tent that filters sunlight. It works great for blocking light and keeping it humid inside to prevent the young leaves from drying out.

Damp Off, Rot around the green stem below the soil line. I pull all my plant up so about a 1/2 inch of root is showing.

One inch of root above soil.

Thank you all so much for your suggestions! I think the problem is too much sunlight and undeveloped roots, as Nik, Alanna, and Kvetch have suggested. I put down the window shade and took them out of direct light, but the leaves did not perk up so I suspected it was overwatering. I reduced my watering but then most of the remaining leaves curled up and dried, so warmer weather and sunlight is most likely the problem. I didn't see any problems with rotting/unhealthy roots when I repotted them.

I will keep them out of direct sunlight and spritz them, as Alanna suggested. Thank you, Kvetch, for the suggestion on how to test watering levels. I already noticed one of my lemons growing a new leaf that looks very healthy, and all the stems still look green so I have hopes that they will bounce back. I will post again after they seem to be healthy again, to say what worked for sure.

I certainly don't agree with growing new young seedlings in shade. Citrus nurseries grow new seedlings in the sun. Dr. Carl E., Whitcomb, Ph.D. author of "Plant Production In Containers-11", owner of Root Maker Container Corp, and long time professor of horticulture further advises to put new seedlings outside in some wind starting when the seedlings are 2" tall to flick the little trunks back and forth thus developing strong hardy trunks. We have followed this course for years with near zero failures. Growing seedlings in the shade is asking for trouble.

Silica the advice you quoted focuses on wind and building trunk strength. I agree that is definitely something to provide growing seedlings. The advice makes no mention of being in direct sunlight.

Shade might not be the word to use. Direct light but filtered might be what I mean. I don't believe 2" seedlings should be under full direct sun 6-8 hours. When I had my seedlings in my bedroom window facing south, they received a lot of direct light from the sunrise and sunset. They received no direct light around the noon hours since the sunlight's angle would be directly over the roof and blocked from entering the window. They did not get dehydrated this way. Now that they're outside, one of my seedlings got sunburned on its green trunk because I lifted the flap to give them more direct noon sunlight. I keep the flap closed now to prevent anymore sunburns from the noon sun

All of our seedlings, mostly Sour Orange, and various root stock varieties, are grown in direct full sun from their emergence from the soil forward. Seedlings have no trouble with sun. It is a common practice in the nursery business.

Thank you all for your clarifications!

I'm still worried about putting the seedlings in full sunlight, because I think many of the leaves were sunburned/dehydrated when I kept them outdoors. I don't have them in complete shade—my window blinds are a thin fabric, so I guess I would describe it as filtered light, as Kvetch said.

I'm also curious about Silica said about keeping them outside to let the wind build trunk strength. I feel like the seedlings are still too small to keep them outside, especially because the wind is quite strong here in the Bay Area. By wind, do you mean enough wind for them to sway very slightly from side to side? When I keep them outside, they are almost blown completely over.

I don't have that many plants and they're all on concrete floor in my south facing backyard. I don't know what a nursery's seedling raising space is like but mine is too dry, hot, and reflective of an environment for my seedlings. I only mentioned one seedling being sunburned bc that was the one to survive but a few others withered away from dehydrating even when they were watered. Perhaps I don't have them potted in large enough containers to keep the temperatures down like a nursery might. It's good to know direct sunlight works on seedlings so I may try putting a few that I don't want in the direct sun

I put my seedlings in direct sun and make sure they are watered properly. Never had a problem with damp off (and I water from the top).

One caution, if a tree has been babied and grown inside in low light then the tree will have to be slowly adapted to the sun. However, if the tree was born in the sun light, then it will be ok. In nature seedling citrus trees have started from day one in direct sun for the last millions and millions of years.

Yep that's the culprit! I was thinking if the seeds are sown and set in direct sunlight to sprout then most likely it wouldn't have any problems with it. Most backyard hobbyists will grow and sprout the seeds indoors until a certain size before deciding to put it in direct sunlight. That's what I did! It's probably a habit from learning to sow vegetable seeds indoors and transplanting them outdoors later when they're larger.

'..... I didn't change anything in how I took care of these plants,...'

And yet you placed them in full sun.

'...I put them outside in sunlight, but it was too hot and dry so some of the leaves shriveled up....'

Bingo. They are sun burned. It wasn't that it was too hot or dry. Many citrus grow in hot, dry places. But plants that have been growing indoors have leaves adapted to lower light levels. It takes about 3 weeks to slowly adapt them to full sun outdoors.

Key Lime from seed in smart Parrot Pot. Under LED light. I cycle water between 38-50%.

I place my seedlings under the canopy of my larger container trees, dappled sunlight and wind makes them strong and also watching the watering. They sit on top of the soil of my larger container trees outside in their small individual pots, some of them I sprouted directly in the soil in their own small containers, outside under the larger trees, haven't lost any seedlings.

Try to not use transparent cups for any type of plant or seed, roots should be growing in the dark.