Classification problem where one attribute is a vector
Yes.  If you have a non-trivial data set of this form, this would be a reasonable fit for statistical analysis.

For independent variables, you have one binary feature for each core item (indicating whether that item was purchased or not); the outcome is a binary variable (win or loss).  Accordingly, one reasonable approach would be to try logistic regression.  You'll have one independent variable $X_i$ for each item; $X_i=1$ means that item $i$ is one of the 6 items that the champion purchased in this game, $X_i=0$ means item $i$ was not purchased.  You'll have a dependent variable $Y$; $Y=1$ means that the champion won this game, $Y=0$ means the champion lost.  Then logistic regression will tell you which items tend to be associated with winning games.

There are other methods you could try as well: pretty much any method for supervised classification that works well with binary/categorical variables.

The one thing I don't recommend you do is multiply the item codes.  That's not going to help.  Instead, just have 20 features, where each feature indicates whether a particular item was purchased or wasn't.