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This is not quite what you mean, but relevant. Remember that a strictly associative and unital symmetric monoidal category is called a permutative category. I observed ages ago (http://www.math.uchicago.edu/~may/PAPERS/13.pdf) that the unbiased version of that is the same thing as an algebra over the categorical Barratt-Eccles operad. Recently Corner and Gurski, http://front.math.ucdavis.edu/1312.5910https://arxiv.org/abs/1312.5910, defined pseudoalgebras over operads in $Cat$. A pseudoalgebra over the categorical Barratt-Eccles operad is the unbiased version of a symmetric monoidal category that you are looking for, except that you ask for functors defined on all unordered finite sets. But, as usual, it is standard and reasonable when considering diagram categories, to pass to a skeleton so as to replace essentially small domain categories with actual small ones, and then one arrives at the definition just cited.

Incidentally, this unbiased variant is essential in equivariant theory in progress where we define genuine symmetric monoidal $G$-categories and show how to construct genuine $G$-spectra from them, where $G$ is a finite group (joint work with Guillou, Merling, and Osorno, to be posted soon; see http://www.math.uchicago.edu/~may/TALKS/Chicago2015.pdf).

This is not quite what you mean, but relevant. Remember that a strictly associative and unital symmetric monoidal category is called a permutative category. I observed ages ago (http://www.math.uchicago.edu/~may/PAPERS/13.pdf) that the unbiased version of that is the same thing as an algebra over the categorical Barratt-Eccles operad. Recently Corner and Gurski, http://front.math.ucdavis.edu/1312.5910, defined pseudoalgebras over operads in $Cat$. A pseudoalgebra over the categorical Barratt-Eccles operad is the unbiased version of a symmetric monoidal category that you are looking for, except that you ask for functors defined on all unordered finite sets. But, as usual, it is standard and reasonable when considering diagram categories, to pass to a skeleton so as to replace essentially small domain categories with actual small ones, and then one arrives at the definition just cited.

Incidentally, this unbiased variant is essential in equivariant theory in progress where we define genuine symmetric monoidal $G$-categories and show how to construct genuine $G$-spectra from them, where $G$ is a finite group (joint work with Guillou, Merling, and Osorno, to be posted soon; see http://www.math.uchicago.edu/~may/TALKS/Chicago2015.pdf).

This is not quite what you mean, but relevant. Remember that a strictly associative and unital symmetric monoidal category is called a permutative category. I observed ages ago (http://www.math.uchicago.edu/~may/PAPERS/13.pdf) that the unbiased version of that is the same thing as an algebra over the categorical Barratt-Eccles operad. Recently Corner and Gurski, https://arxiv.org/abs/1312.5910, defined pseudoalgebras over operads in $Cat$. A pseudoalgebra over the categorical Barratt-Eccles operad is the unbiased version of a symmetric monoidal category that you are looking for, except that you ask for functors defined on all unordered finite sets. But, as usual, it is standard and reasonable when considering diagram categories, to pass to a skeleton so as to replace essentially small domain categories with actual small ones, and then one arrives at the definition just cited.

Incidentally, this unbiased variant is essential in equivariant theory in progress where we define genuine symmetric monoidal $G$-categories and show how to construct genuine $G$-spectra from them, where $G$ is a finite group (joint work with Guillou, Merling, and Osorno, to be posted soon; see http://www.math.uchicago.edu/~may/TALKS/Chicago2015.pdf).

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Peter May
Peter May
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This is not quite what you mean, but relevant. Remember that a strictly associative and unital symmetric monoidal category is called a permutative category. I observed ages ago (http://www.math.uchicago.edu/~may/PAPERS/13.pdf) that the unbiased version of that is the same thing as an algebra over the categorical Barratt-Eccles operad. Recently Corner and Gurski, http://front.math.ucdavis.edu/1312.5910, defined pseudoalgebras over operads in $Cat$. A pseudoalgebra over the categorical Barratt-Eccles operad is the unbiased version of a symmetric monoidal category that you are looking for, except that you ask for functors defined on all unordered finite sets. But, as usual, it is standard and reasonable when considering diagram categories, to pass to a skeleton so as to replace essentially small domain categories with actual small ones, and then one arrives at the definition just cited.

Incidentally, this unbiased variant is essential in equivariant theory in progress where we define genuine symmetric monoidal $G$-categories and show how to construct genuine $G$-spectra from them, where $G$ is a finite group (joint work with Guillou, Merling, and Osorno, to be posted soon; see http://www.math.uchicago.edu/~may/TALKS/Chicago2015.pdf).