Kent Police have arrested a 53-year-old man from Chatham on suspicion of sending the package. He remains in custody as enquiries continue.
The package is understood to have arrived at the production plant in Wrexham on Wednesday morning.
Police said pre-planned warrants took place at addresses in Luton Road and Chatham Hill on Thursday morning.
A spokesperson for Kent Police said: "A 53-year-old man from Chatham has been arrested on suspicion of sending the packages and remains in custody as enquiries continue.
"There is no evidence to suggest there is an ongoing threat."
The factory, which manufactures the AstraZeneca coronavirus vaccine, was partially evacuated on Wednesday following reports of a suspicious package.
Bomb disposal experts were in attendance and police set up a security cordon around the site, closing roads to all traffic.
The Wockhardt factory, which is housed in the Wrexham Industrial Estate, has been said to be able to produce an annual total of 300 million doses of the vaccine created at Oxford University.
North Wales Police reopened roads near the site just before 4.30pm on Wednesday, and workers were allowed back in to the plant. There were no reported injuries.
Wockhardt UK said in a statement: "This temporary suspension of manufacturing has in no way affected our production schedule and we are grateful to the authorities and experts for their swift response and resolution of the incident.”
The incident comes amid mounting tensions between AstraZeneca and the European Union over delays in the delivery of agreed doses.
Addressing rising anger in the bloc over delays to its supply of jabs, AstraZeneca’s chief executive Pascal Soriot said that “glitches” in supply to the bloc were the result of Brussels taking three months longer than London to sign an agreement with the company.
He confirmed that the UK would have the first claim on vaccines produced in the country. AstraZeneca is running two months behind schedule due to production issues at plants in the Netherlands and Belgium.