I don't believe there is anything that would prohibit a road association from posting its roads in mud season. Enforcement is another matter, however. There is very little that law enforcement agencies can do when it comes to private roads.
One thing you could try is amending your bylaws to say that if anyone causes damage to the road beyond normal wear and tear, they may be billed for repairing the damage. But once again, there is little you could do to enforce such payment if a person contests what constitutes normal wear and tear during mud season.
I do know of one private road that posts a series of signs during the summer reminding people to slow down so as not to raise dust. (That dust consists of the "fines" which are the glue that holds a road together.) You could try posting a series of signs reminding people to please be considerate, that all of you depend on the road for access, and that damaging the road raises maintenance costs.
Even the official town postings allow fuel deliveries on posted roads, but it certainly makes sense to plan ahead to avoid that kind of burden on roads that are not built to town road standards. It could be helpful to send out a notice before mud season, reminding people to fill their fuel tanks and schedule any other heavy delivery trucks to make their deliveries before the road thaws, or to arrange to have small package deliveries made elsewhere for pickup until the road firms up again.
On the other hand, if your road has been designated as a public easement, there may be some assistance ahead. LD 1985 has passed both the House and the Senate. Hopefully it will also get approved by the Appropriations committee. If it does, and becomes law, it will contain a provision making it clear that towns have the authority to post public easements in mud season, the same as they do with town ways. You could ask the town to post your private road, but they are not obligated to do so.