A New Jersey couple was heading home from dinner in an Uber in March 2022 when their driver T-boned another car, leaving them with serious injuries, including spine and rib fractures.

The couple, Georgia and John McGinty, of Princeton, N.J., sued Uber nearly a year later. Now, their effort to bring the case to court could be hampered by a terms-of-service agreement that they say their 12-year-old daughter signed while ordering pizza using Ms. McGinty’s Uber Eats account.

A New Jersey appeals court found last month that the agreement’s arbitration provision — which says that most disputes between Uber and its customers must be settled privately — was “valid and enforceable,” reversing a lower court’s decision that would have allowed the couple’s personal-injury lawsuit to be heard by a jury.

The car crash left the McGintys severely injured. Ms. McGinty, 51, had cervical and lumbar spine fractures, rib fractures, a protruding hernia and other injuries. She had numerous surgeries and was unable to work for more than a year, until April 2023. Mr. McGinty, 58, suffered a fractured sternum and severe fractures in his left arm and wrist, and has not regained full use of his wrist.

“This happening was like a bomb going off in our life,” Ms. McGinty said, adding that the crash was devastating to “our health, our emotional well-being, our financial well-being” and “our ability to parent our young child.”

Mr. McGinty said, “We’re in constant pain every day.”

They are seeking damages, though their lawyers declined to specify an amount.