Article:21 September 2015 Last updated at 10:58 BST
Figures from the body show Coventry City Council is among the authorities falling behind.
A total of 7,230 homes were built in Coventry between 2011 and 2014 - a 60% shortfall on the calculated need.
Councillor Kevin Maton, cabinet member for business, said the authority needed 40,000 new homes over the next 15 years.
Reece Palmer and his girlfriend Ayesha Kalyan are among those affected.
"If the prices are at a certain point now, who knows what they're going to be like in a year or so?" said Mr Palmer.
Inside Out is on BBC One West Midlands on Monday, 21 September at 19:30 BST and nationwide on the iPlayer for 30 days thereafter.

A summary of the above article is?
Young people in the West Midlands are struggling to get on the property ladder because councils are failing to meet house building targets, according to the National Housing Federation.