Can third-party groups design Wisconsin’s ballots?

No.  Wisconsin State Statute 7.10 explicitly gives the responsibility of preparing, printing, and distributing ballots to county clerks, based on templates developed and approved by the Wisconsin Elections Commission for each election cycle.  

Ballot templates are approved by the Commission at a public meeting, which gives the public, political parties and elected officials an opportunity to provide feedback in advance.  Once approved, ballot templates must be used by county clerks to create the ballots for each municipality.  County clerks provide those ballot designs to WEC staff for an additional review, and only slight variations from the template are allowed.
County clerks design and lay out ballots in conjunction with the programming of electronic voting equipment.  Many county clerks do this work in their offices with their own staff and computers. Others have contractual relationships with voting equipment vendors to provide these services.
County clerks are also responsible for arranging for the printing all the ballots for their municipalities.  Some clerks print ballots in-house, while others use vendors who specialize in ballot printing.  Once printed, county clerks are also responsible for delivering ballots to the municipal clerks by deadlines established in state statutes so they can be mailed to absentee voters.  

Attached is a document that explains in greater details the process flow for how ballots are designed and printed in Wisconsin.