Many people I know travel over the state border here to purchase things, especially alcoholic beverages, since the nearby state's sales tax is much lower. Are there moral objections to avoiding taxes in cases like this?

Let me state at the outset that I am not a civil lawyer. But let us suppose that there are legitimate laws in place that forbid purchasing and carrying certain items across state lines. Presuming such laws do exist, they ought not to be violated. This would be true even if we suppose that such laws are more likely aimed at large distributors, rather than private citizens. Unless that exception is explicitly indicated in the law or overruled by other constitutional rights, we ought not to simply consider ourselves exempt. 
 
It is a general norm for Catholics that we should obey all just laws enacted by civil authorities if they do not violate God's laws. If such laws displease us, we ought to have recourse, through the political process, to change them, rather than simply ignore them. 
 
If a Catholic knowingly violates just civil laws such as this, the gravity of such a sin varies, based on the frequency, purpose and amount of items purchased. If one were to simply purchase a small number of items, more as a convenience, the gravity of such sin would be rather light. However, if one were to repeatedly purchase a large amount of items, especially if they were going to resell them, this would be a much more egregious violation of civil law, and also bring with it a more significant gravity in violation of moral law related to the 4th Commandment.