Use of commas when naming multiple family members

Using semi-colons would not be incorrect here but there's nothing wrong with the way you have it, using just commas.

In American style as it has developed over the last few decades, the trend is to use fewer commas and other punctuation marks. In that style, you would certainly dispense with the next to last comma. In fact, you might write:

"I live with my mother June Smith, my brother Robert Smith, my sister Jenna Smith and my cousin Avery Johnson."

But suit yourself. Please leave semi-colons alone unless you have an independent clause followed by a dependent clause, or two independent clauses in the same sentence.