According to Conrad Stibbe in his excellent book Das Andere Sparta no less than nine Lacedaemonian artists are known to have worked in Olympia alone. While the majority of these artists are described as Lacedaemonian, in two cases, Syadras and Chartas, the artists are explicitly referred to as Spartiate. While it is possible these were the only exceptions in Spartan history, it is more likely that they are the tip of the iceberg: the only surviving record over two and a half millennia of other nameless Spartiate artists.
Meanwhile, we should not ignore the plethora of smaller art objects from bronze vessels and jewellery to statuettes and figurines found at Spartan sanctuaries which record a thriving industry for domestic craftsmanship if not high art. These are well catelogued by Reinhard Foertsch in his article "Spartan Art: It's Many Deaths," in Sparta in Laconia: Proceedings of the 19th British Museum Classical Colloquium, Dec. 1995 (Cavanagh, WG and S.E.C. Walker, eds.) The same publication contains an excellent article by Maria Pipili, "Archaic Laconian Vase-Painting," which highlights the sophistication and high quality of 6th Century Laconian pottery.
Altogether, archeological research suggests that art was more common and more valued in Sparta than is widely acknowledged today. Spartiates certainly bought works of art and dedicated art objects at their sanctuaries. The extent to which they engaged in production of art themselves will never be known but, as noted above with respect to the two sculptors, at least in some cases Spartans were professional artists.