The new 9,100m2 college building is scheduled to be completed in autumn 2015.
The client project team includes Buro Happold as building services consultant, structural engineer Hydrock and Gleeds as cost consultant.
Funded principally by an award from the Skills Funding Agency, the £21m project forms part of the local economic partnership (LEP) initiative and Chippenham Vision Master Plan.
Willmott Dixon's main construction contract is worth £15.3m.
The Chippenham campus – one of four operated by Wiltshire College – will offer vocational courses including construction and engineering.