Assuming that there's no legal issues (like the slave doesn't mind, or doesn't know that he can run away), can one buy a slave nowadays in America to have the laws of an Eved Kenaani?
Let's tweak the question a bit. Suppose a non-Jewish woman wants to convert, and meanwhile there is a Jewish mamzer who would like to get married. Can she do a quasi-conversion such that theoretically her status would be "Canaanite maidservant", at which point she'd be allowed to marry the mamzer?
Rabbi Aaron Soloveichik insisted vociferously that the US prohibition on slavery would void any process, however theoretical, that would wind up with someone in a category called "slave." I believe that there are those who differed, but the practical conclusion is generally "no."
No. The sale needs to be valid according to American law as well as per the halacha of Dina Demalchusa Dina - Secular Law. And clearly the Thirteenth Amendment in the constitution forbids the institution of slavery.
This principle can be seen from the Chazon Ish (שביעית כד). The Chazon writes, among several objection to institution of Heter Mechira in Israel, that since the sale to the Gentile is not recorded in the land authority registry (Tabo) the title is not legally transferred and the sale is not legitimate. Since the sale is not legally binding according to the laws that govern such transactions.
His conclusion applies here equally as it is not an effective sale according to state law.