There is no specific regulation addressing "open shoes in the veterinary setting" in any OSHA standard. The rules just don’t don't get that specific because they want to allow the leadership some flexibility in making those decisions.
However, OSHA does have a "foot protection" standard (29CFR1910.136) that reads :
"The employer shall ensure that each affected employee wears protective footwear when working in areas where there is a danger of foot injuries due to falling or rolling objects or objects piercing the sole..."
The determination of whether a hazard exists in the workplace is up to the employer; however, in making this determination, the employer must use logic, knowledge about the hazards in the workplace and industry guidelines from professional or trade organizations.
So, what’s our recommendation? We recommend that open toed shoes be prohibited in the veterinary practice setting. It's our considered opinion that there are too many things that can fall off counters which will injure an unprotected foot. And any type of shoe with openings in the foot covering, would just not provide even the most miniscule amount of protection. Yes, some closed toed footwear such as canvas sneakers offer minimal protection from these hazards, but we think minimal protection is better than no protection.
Of course, for large animal practices, because of the danger from the patients, even canvas sneakers may not be enough. Sturdy boots, even safety-toed boots are the normal and just simply a good idea.
It’s also our advice to avoid shoes that have little or no back to keep them on the foot. When a person trips, if the shoe is not securely attached to the foot, it provides absolutely no support to the ankle and may even make the situation worse by coming off only partially. In those cases a stumble may turn into a fall when the half-on, half-off shoe prevents the staff member from regaining their balance properly.
In 2019, 880 workers died in falls, and 244,000 were injured badly enough to require days off of work. A worker doesn't have fall from a high level to suffer fatal injuries; 146 workers were killed in falls on the same level in 2019.
Source: National Safety Council
https://www.nsc.org/work-safety/safety-topics/slips-trips-falls