It sounds like too little light, although the lying down part is a little odd, unless they're at least around 4 weeks old and starting to fill out and get top-heavy.
Are these guys planted in pots indoors, or in the ground in the garden? If it's outdoors, that's quite weird, unless there's little light where they are.
Just about everything started indoors under fluorescent or window light develops quite a bit of stem, even when the same veggies seem to be more or less stemless when started in the field.
You can try more light, plus motion treatment, blowing 'em with a fan, or brushing them with the edge of a piece of paper or even your hand to encourage stockier growth. But once they've stretched and flopped, it's too late for that.
Too shallow I'm not sure of. Most of the time, most veggie seeds seem to burrow down fine, although once in a while, I'll have a just emerged seedling that germinated right on the surface and is growing kinda on the surface then turning up, with a bit of the root exposed. I'll either pull them, or bury them deeper.
From annoying experience, I've transplanted my share of stretched seedlings: if they're healthy, most veggies will grow OK, the stems will just fill out and bend up to support the rest. You can also bury some of the stem, that generally doesn't hurt 'em, although it's only tomatoes (of the usual garden veggies) that you can really bury and the stems will develop roots.
That's what I've got... Hope it helps!!