A fungus enjoying the damp conditions. Unlikely to be a problem for your seedlings except as a symptom of possible overwatering. But what might be a problem is that egg cartons are not really large enough to raise a plant to transplant size. Were you planning to pot them on?

Yes, originally the plan was to do container gardening, but my landlord provided us with land and soil to make a big garden. The egg cartons I am only using for getting the seeds started then transferring them to a bigger container until it is time to harden off and plant outdoors (which I don't want to do yet because we had snow last weekend and frost warnings earlier this week). I was thinking it was just a fungus due to the egg cartons holding more moisture than the plastic seed trays do but figured I had better check. This is my first time growing anything lol so any info/advice is appreciated :+)

I should mention that everything I have planted in the egg cartons are only about an inch high so far. I am transplanting them to a bigger container this weekend. But most of my other veggies have already been transplanted to large containers already.

I think you should sow the seed direct in ground.
Suzi