The facility was set up by the Scottish Government three years ago to serve the whole of Lanarkshire, at a cost of around £100,000 a year.
However, this cash is set to run out in June and North and South Lanarkshire Councils have received funding applications from Scottish Autism to keep it going.
More than 3,500 people have signed a petition started by Karen Noble, whose daughter Amy attends the One Stop Shop, pleading for money to be found to let it to continue.
On Monday Councillor Bonnar met a number of families who use the service and he has written to North Lanarkshire Council leader Jim Logue requesting a resolution.
He said: “I was in a meeting with parents and service users of the One Stop Shop in Motherwell, hearing first hand of the vital services that this facility provides and the support it offers to schools across North Lanarkshire too.
“Some of the stories behind each family really hit home about the need for this facility which supports the families.
“To stand in a room as a councillor and see parents cry due to the threat of closure and an end to this lifeblood service was very humbling.
“I have urged the leader of the council here in North Lanarkshire to take action fast to protect these vulnerable children, young persons and their parents and carers.”
Both councils have pledged to continue supporting people with autism, but haven’t yet decided if this should be through direct funding of the One Stop Shop.