Yes, you can. There is a type of tumour called a teratoma (which literally means ‘monsterous tumour’), which can occur. Sometime a women gets pregnant and instead of a baby growing, a tumour grows instead, feeding off the mother via the placenta. At other times, a completely normal-looking baby can be born, with a teratoma inside it or attached to it (usually attached to the buttocks).
These types of tumours are generally quite different from other cancers. They can grow teeth and hair. Some have muscles and a nervous response, so they will flinch if you poke them. On rare occasions, there can be a foot or an arm growing out of them and some have eyes. I saw a photo of one, which grew inside a childs brain, with a fully formed hand. There has even been a case, in Turkey, where one had a beating heart.
So yes, you can be born with cancer, and that particular type of cancer is, indeed, monsterous.
There are extremely rare cases of newborn cancers (I found newborn neuroblastoma), but as I said, they are extremely rare.
More frequently, you could be born with a predisposition to cancer, that means having some mutations in genes that are associated to develop some forms of cancer (for example, mutations in the genes BRCA1 and 2 are associate to breast cancer) . However, always remember that having these mutated genes give you a predisposition, but not the certainty that you will develop cancer, because multiple genes have to mutate to transform a cell into a cancer cell.