The move follows confusion and chaotic scenes at the borough's assessment centre off Jiggers Bank near Ironbridge on Tuesday which saw hundreds of cars arrive after being sent there by the NHS booking system.
Many families who had travelled from across the country for appointments were turned away after staff ran out of test kits.
In the letter, council leader Shaun Davies and health boss Councillor Andy Burford stated: "National contact tracing is not working – it is failing to reach cases and contacts sufficiently and not able to identify outbreaks early enough.
"Our approach in Telford and Wrekin is to undertake contact tracing ourselves so that we can understand more fully transmission of the virus within the borough and be best placed to take swift action to contain and stop further spread.
"We are quite happy to manage and provide the whole contact tracing function for our borough but we would ask that this is properly resourced through transfer of cash or staff from NHS Test & Trace.
"The resources spent on it have been huge in scale compared to those allocated to local government.
"The system has been designed centrally and is clearly not working.
"Our council is doing all it can to prevent and contain the spread of the virus but within a poorly designed national system – I urge you to resolve these issues quickly as the potential health, wellbeing and economic impacts risk creating an autumn surge that could irreparably damage public trust in and compliance with government advice."
The Department of Health and Social Care initially denied there had been system glitches at the Telford centre, but Health Secretary Matt Hancock has since pledged to find the cause.
A department statement said: “NHS Test and Trace is completely free, it’s working and our capacity is the highest it has ever been with over a million tests being processed a week.
“We are seeing a significant demand for tests and so those attending a regional test site or mobile test site without an appointment may be advised to book before they are able to receive a test. If you have symptoms we urge you to visit the NHS website or call 119 to book a testing appointment.”
The borough's public health director Liz Noakes said: “You should not get a test if you only have symptoms such as a sore throat or runny nose. If you have been in contact with someone who has symptoms then you should immediately self-isolate – only get a test once you develop symptoms.
“People may be turned away from the Ironbridge Park and Ride if they turn up without an appointment or do not have coronavirus symptoms.”