The 85-year-old had an altercation with the suspected robber at about 11.30am on Friday but escaped.
She got into her car on Whitchurch Road in Wellington but the alleged thief approached, opened the door and grabbed the bag.
A 29-year-old woman has been arrested on suspicion of robbery, and police are appealing for help.
Detective Constable Lizzi Waterworth of the Telford Criminal Investigation Department said: "Our detectives are working hard to identify the person responsible and we are following a number of lines of enquiry.
"There were a lot of people in the area at the time of the incident so we're keen to hear from anyone who witnessed it or has any other information that could help with our enquiries."
Anyone who saw what happened or has any information is asked to call 101 and quote incident 216s of January 11, or the independent charity Crimestoppers anonymously on 0800 555 111.