Unless said state has a specific law against traveling out of state to get an abortion, they could not legally charge someone who did so. As of yet, I haven't heard of any state seriously considering one.
This could change, but I think it would turn out badly for the state that does it. For the most part, people that would travel out of state to get an abortion would be those that could afford to do so. Remove that option and many will choose to relocate out of state. This would drain the state of a portion of of its high income earners, making it a "poorer" state, and less attractive to new businesses, starting in a downhill slide.
Then there is the fact that a good number of legislators who vote for anti-abortion laws in their state don't do so from their own moral position, but because it panders to a specific voter base. They are in that group that can afford to, and would have out of state abortions done. They would be reluctant to close that loophole.