YES.
Synthetic DNA fragments containing promoter sequences can be used as template DNA. In addition, template DNA can be prepared from synthetic DNA of ORFs by PCR using primers that include the promoter sequence and other necessary sequences.
However, depending on the manufacturer of the gene synthesis, the delivered product may contain RNase, which inhibits the protein synthesis reaction. If the protein of interest cannot be synthesized, try adding an RNase inhibitor or inactivating the RNase.
Also, if the product is delivered as a plasmid, please check the notes for using the plasmid as a template DNA.