Amesbury-based Mears Care Limited was criticised in a Care Quality Commission (CQC) report after it made unannounced visits in June and July.
The report found standards were not being met in five areas including medicine management, staff training and support and quality of service.
Alan Long, from Mears, said it had put in an action plan to improve standards.
Mears took over the contract, from Careline, in May, to look after the elderly in their homes.
Mr Long said an unexpected number of staff then left, leaving Mears "unable to operate properly".
He said the company "sincerely apologised" to the service users and their families who had seen "their service deteriorate since May".
He said: "The office staff left without notice and the majority of care staff left with them, leaving us with very poor cover."
Adrian Hire, whose 92-year-old mother died last month, said he could "not help wondering" if his mother "may have lived longer" if Mears had not been responsible for her care.
"Her health slowly diminished when Mears got involved," he said.
Councillor Christine Crisp, from Wiltshire Council, said they were looking "very carefully" at the action plan submitted by Mears.
She said: "What Mears have to do now is answer the criticisms of the CQC. The CQC will have to re-inspect to make sure Mears have lived up to their promises.
"At the same time, Wiltshire Council will be looking to ensure that our patients and people requiring care are getting it."
A home care firm in Wiltshire has apologised to clients and their families for its poor service.