Officers from the force’s Roads Policing Unit posted a picture of a man and woman carrying their luggage after being stopped on Horbury Road in Wakefield, at around noon on Thursday.
The tweet, which has since been removed from the unit’s account, included the hashtag #luggagewalkofshame.
West Yorkshire Police apologised and a spokesman said it was investigating why the tweet was posted.
He said: “This was clearly an inappropriate tweet which we immediately removed from our Twitter account, and we are looking into the circumstances as to why it has been posted.
“Regardless of any offences people have committed they still deserve their human dignity and we would apologise for the offence this has caused.”
But Twitter users said the tweet was “absolutely disgusting and claimed the officers should have offered the couple help.
Ami Slaven tweeted: “disgusting behaviour... What happened to you protecting and serving the needs of the community?”
Adam W tweeted: “I do hope you put them in touch with organisations that can help them in their challenging situation? Disappointed by tweet.”
And Holly Parry tweeted: “Car insurance is the least of their worries. This tweet shows total lack of respect from police #disgraceful.”
Kate Webb, deputy head of policy at Shelter, said: “The fact that people are driven to situations like living out of their cars is yet another terrible example of how far ordinary families are being pushed, and the inability of the housing safety net to support all those who need it.
“With housing costs sky-high, and the drastic shortage of genuinely affordable homes never more apparent, we are hearing from increasing numbers of families who are terrified that just one thing, like a sudden illness or job loss, will leave them homeless.
“We would urge anyone worried about losing their homes to come to us for expert advice as soon as they start having problems paying their rent or mortgage. It could make the difference between losing your home and keeping it, and ensure that support is there if the worst does happen.”