Visit the Butterfly Gardening Forum for advice about raising caterpillars. However, caterpillars often die in captivity, especially if under the care of a novice. It's not something that I would attempt myself. I let mother nature handle things.

But, you'll get some ideas in the other forum. I wish you good luck.

Note that when asking for advice on the above topic, "Yellow swallowtail" is not a butterfly name in common use. Perhaps you are referring to the mainly yellow Tiger Swallowtail, but that butterfly does not use carrots as a host plant for the caterpillars. There are other swallowtails that use parsley family plants (like carrots).

Most people on the butterfly forum are using various inexpensive laundry container setups, or they build their own cage out of wood and screen. An aquarium would work if covered with a piece of screen and kept out of direct sun.

Posting on the butterfly forum is a good idea for more detailed advice.

...and that is what mpoland has done.