It'll be dependent on how they frame the law. Medical stuff can be a bit odd, especially when it comes to getting care in other states. A good example came to light during the pandemic with telehealth. Unless a physician is licensed to practice medicine in a given state, they can't provide telehealth there. To give a practical example, when I traveled back to Michigan I had to reschedule a virtual appointment since none of our doctors are licensed to practice medicine there.
I'm just spitballing here, but I suspect Missouri could use that to make it illegal to get an abortion across state lines by defining what constitutes "care". So say you live in Missouri and call a clinic in Illinois to get an appointment to have an abortion. Depending on how Missouri defines care, they could say that merely calling a clinic counts as "medical advice" thus subject to having physicians that need to be licensed in that state.
They couldn't go after the patient, but they could certainly make a fuss with the state licensing board.
Or they could phase it in a way to say that someone going to another state for a procedure is a form of tax aversion. I think that's how Utah can technically make it illegal for me to go to Wyoming to buy alcohol and then transport it back into the state.
They'd still have to prove it though and since HIPAA is a thing, it's not like the states could easily get medical records without jumping through hoops. I believe to access them, they'd need a warrant and interstate warrants are a tricky thing.