You are absolutely going to have to buy at least one seat for one of the children. I don't know if you will have to buy seats for both.

You might want to seriously reconsider taking this trip at another date when you have someone with you to help. I don't know how one person can safely handle 2 children under age 2. At the very least, you will have to rely on a lot of help from strangers, which might not be very forthcoming.