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I have a data set where many of the actual values are zero, so I can't use MAPE. It's not a time series, so I can't use MASE ala our very own Rob Hyndman.

Is there another alternative to MAPE that I could use?

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MASE is suitable for non-time series data also. See my textbook: https://www.otexts.org/fpp/2/5

In that case, you can scale the data using the mean as the base forecast. So if $e_j$ denotes a prediction error on the test data, then the scaled errors are $$ q_{j} = \frac{\displaystyle e_{j}}{\displaystyle\frac{1}{N}\sum_{i=1}^N |y_i-\bar{y}|}. $$ where $y_1,\dots,y_N$ denotes the training data.

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