Morton has stepped down due to "health issues", the Newgate Communications, PPS Group, Publicasity, Redleaf and Thirteen owner said.
PRCA chair Williams joined Porta from Finsbury last year as group MD. He succeeds Wright as CEO.
Meanwhile, non-executive director John Foley has become senior independent director.
The changes take place with immediate effect.
Wright, who created Porta in 2010 having previously co-founded Citigate, said: "I am pleased to announce that Steffan has taken on the CEO role to lead the next chapter of Porta’s development, supported by the board and myself as chairman.
"I would also like to thank Bob Morton for his very considerable contribution as chairman to date. It’s no exaggeration to say that without Bob’s support, Porta would not have achieved the rapid growth shown to date."
Earlier this year, MHP chief executive Gavin Devine joined Porta in the new role of chief operating officer.
Last month Porta reported a dip in fee income in the first half of 2016, pointing to the negative impact of the EU referendum on its UK business.