Despite the ubiquity of multiway data across scientific domains, there are few user-friendly tools that fit tailored nonnegative tensor factorizations. Researchers may use gradient-based automatic differentiation (which often struggles in nonnegative settings), choose between a limited set of methods with mature implementations, or implement their own model from scratch. As an alternative, we introduce NNEinFact, an einsum-based multiplicative update algorithm that fits any nonnegative tensor factorization expressible as a tensor contraction by minimizing one of many user-specified loss functions (including the $(α,β)$-divergence). To use NNEinFact, the researcher simply specifies their model with a string. NNEinFact converges to a stationary point of the loss, supports missing data, and fits to tensors with hundreds of millions of entries in seconds. Empirically, NNEinFact fits custom models which outperform standard ones in heldout prediction tasks on real-world tensor data by over $37\%$ and attains less than half the test loss of gradient-based methods while converging up to 90 times faster.