The decision was made on the advice of on-site veterinarians.
The quarantine was in place pending official test results on the animal, understood to be from the Mid West.
Officials will be hoping that tests rule out the neurological form of equine herpesvirus-1 (EHV-1), which is highly contagious among horses and could spark even greater disruption.
The club said additional biosecurity measures have been implemented and exhibitors are co-operating in an effort to mitigate any potential risks of infection.
The club said it was in contact with the Texas Animal Health Commission and was taking its guidance.
It continued: "There is no evidence of any spread of the symptoms found in the original horse, but as part of the precautionary measures, the barns and implements have been sanitised and the process will continue every two hours throughout the day until the voluntary quarantine is lifted.
"The voluntary quarantine is for animals only, and officials have asked that no animals leave or enter the facility."
The club stressed that classes had been cancelled for Wednesday only.
A decision on remaining events will be made pending results of the tests, it said. All other events will occur as scheduled, including the Cowgirl Museum and Hall of Fame induction luncheon.
Kevin Kemp, assistant facilities and public events director, said Will Rogers Equestrian Center staff are prepared for these types of situations and are ready to respond.
"The Will Rogers staff and show organisers are committed to doing everything we can to maintain a healthy and safe environment," Kemp said.
"We are working very closely with show management and are in communication with and are following all recommendations of the show veterinarian and the Texas Animal Health Commission."
Club chief executive Steve Taylor said he is grateful for the co-operation between his staff and the center.
"We appreciate everyone working together on this and for the hard work everyone has put into mitigating any effects from this situation.
"Everything that can be done is being done."
One report said the sick animal was reported to be showing a high fever as well as neurological signs.