Both the Boots and Brown & Francis pharmacies were open but not dispensing drugs but NHS England and NHS Improvement said both were expected to be short term issues, with both opening tomorrow (Thursday).
A spokesperson for NHSE & I said patients for Boots and Brown & Francis were being directed to Lloyds, which remained open for the day.
"Boots is open but not dispensing today due to the pharmacist being ill; they have however secured a pharmacist for Thursday March 31.
"Brown & Francis was open but not dispensing until 1pm today due to the pharmacist being ill, they have however secured pharmacist provision from 1pm and they are now dispensing."
Matt Webster, the head of retail at Peak Pharmacy, which owns 120 pharmacies across England including Brown & Francis, said: "We have owned the Ludlow pharmacy for one year and this is the first time that both ours and Boots has been closed," he said.
"It is a rare event and not a situation we would have wished for."
Mr Webster said if the situation was an emergency one, staff have a way of getting a prescription to another pharmacy.
"We are working hard to get a pharmacist to Ludlow and we hope to rectify this as soon as possible."
Station Drive Surgery patient Mike Fothergill, aged 78, of Caynham, said: "People are running out of vital drugs.
"Not only does it take a long time to get through on the prescription telephone ordering service, but doctors have been taking up to 10 days to sign the prescriptions off. Then if the prescription cannot be taken to a pharmacy, what do we do?"
Mr Fothergill, a diabetic, added: "Somebody ought to provide a way to get hold of drugs. There are certain drugs that people have to take or it gets serious."
On Tuesday, the Shropshire, Telford and Wrekin Integrated Care System (ICS) declared a "system critical incident" covering the county's two main acute hospitals, care homes, GP surgeries and ambulances.
Eight GP practices were operating business continuity plans due to covid staff sickness and both NHS111 and Shropdoc (the county's GP out of hours service) reported staffing issues. Staff on leave have been asked to contact work to see if they are needed.