Law firm Digby Brown said one person was taken to hospital with a fractured shoulder, while other injuries suffered during the “shocking ordeal” included fractures, sprains and soft tissue damage.
It said “dozens more are feared to have suffered injuries” after the Pentland Ferries vessel Alfred grounded on the Isle of Swona.
A total of 45 of the 84 passengers, who included holidaymakers, were evacuated by lifeboat before the 85m-long catamaran was re-floated.
Mark Gibson, a partner at Digby Brown who heads its foreign and travel department, said: “Any mode of travelling has its risks, but there are clear procedures designed to keep people safe, whether there be adverse conditions, mechanical issues or human error.
“For the passengers on board, this must have been a shocking ordeal with the injuries and overall chaos impacting each person in a different way.
“The injuries we’re aware of so far relate to fractures, sprains and soft-tissue damage, but it’s possible there are injuries that are as yet unreported, including those who suffered psychiatric harm.
“Legal proceedings are at an early stage, but we will continue to support and advise anyone affected by this significant collision.”
The firm said the UK Department for Transport’s Marine Accident Investigation Branch (MAIB) investigation into the incident “will likely include a review of vessel operator Pentland Ferries”.
An MAIB spokesperson said: “Our safety investigation is ongoing and a report will be published in due course.
"The MAIB conducts safety investigations intended to prevent future accidents without apportioning blame or determining liability.”
The Royal National Lifeboat Institution said it had assisted with at least four injuries after the grounding.
It said 45 passengers were “extracted” from the ferry by its lifeboat from Longhope on South Walls in Orkney.
A second lifeboat, from Thurso, escorted Alfred, which had a crew of 13, on its return to St Margaret's Hope.
Pentland Ferries said it was unable to comment on the legal claims.
Managing director Helen Inkster had said after the incident: “Unfortunately, the impact of the ship grounding did lead to a few passengers sustaining injuries.
"We understand that the situation onboard the vessel remained calm and we thank everyone for their patience and co-operation in difficult circumstances and very much appreciate all those who came forward to offer their assistance.
“Our first priority is the safety of our passengers and, as such, the crew were quick to implement the safety procedures for which they have been trained.
"Emergency services were called and passengers mustered as per protocol.”
The firm said the ferry's port bulbous bow had been damaged when it ran aground, while its starboard (right) hull and the stern of the ship had remained afloat.
Its spokesperson said: “After confirmation that the ship was not suffering any water ingress, it was floated off unaided and made way to St Margaret’s Hope under its own engine power.”
Alfred entered service in 2019 and can carry 430 passengers and 98 cars.