How do I make curls stay through the day without hairspray?

Try using curling rods.
Put heat on your hair & then put your hair in curling rods.

Do you have any mousse or gel and some velcro hair rollers? If so, just apply some mousse or gel to your hair when it’s damp (not soaking), then put your hair in the rollers and sleep in them. The longer the rollers stay in, the better. Should work pretty well, no hairspray required.

If you don’t have hair rollers, you can also wrap your damp hair around strips of paper bags, and roll up toward your scalp. Then tie the ends of the paper strips together (like the first step of tying a shoe) and sleep in them. Some people don’t require any kind of product when they use this method, just keep in mind the curls are going to be much smaller than they would with rollers. Again, applying gel or mousse would probably make the curls last even longer.

You could also (again, with or without mousse or gel) braid your hair in 6 or 7 pieces while it’s damp, sleep in it, and when you wake up it should be wavy.

What products do you have?

The old-fashioned way ( See Nellie Olson on Little House on the Prairie) is to dampen hair and use strips of clean cotton rags as curlers. They don’t hurt too much when you sleep on them.

Personally I never heard of using strips of paper bags but I spent my youth trying to get rid of my curls and waves.

The OP doesn’t say she can’t curl her hair, she is asking how to make the curls stay.

My suggestion is use plenty of mousse or gel before you curl.

Pinning them up with bobby pins and letting it sit for 5–10 minutes after you curl them seems to make mine hold the shape a little longer.

I use aloe vera gel instead of traditional hair products and my curls stay.