I'm not an attorney, but I see nothing in the Road Association Statutes (23 MRS 3101-3104) that would specifically prohibit closing a private road.  In theory, if the Members vote to close it, or if they vote to put a provision in their bylaws saying they can close it, there would be nothing within those statutes to prohibit it. 

HOWEVER, if it's the only access to a year round residence, that would be problematic.   There's a Maine Supreme Court case, Jordan v Canton, that says that access is a property right, attached to property, and that access cannot be taken or damaged without both due process and just compensation.  Although the case is referring to governmental actions that damage property access, the same principle should apply to road associations, which act sort of as a mini-governmental body. 

If there is another access to each property, closing a road during mud season shouldn't be a problem.  Or if the landowners/residents affected want it closed to keep others from using it, that also should be okay.  If it gives access only to seasonal camps, it could also work.  But if you are going to prevent even one person from reaching their home by vehicle for four to six weeks while the road dries out, and that person doesn't agree, that IS a problem. 

I would suggest that what you could do instead is to add a provision to your bylaws (if you don't already have it) that says if a member damages the road beyond normal wear and tear, that person is responsible for paying for the repairs.  Then the question is, does using the road for access to one's home during mud season qualify as "normal wear and tear?"  That might be a judgment call, i.e. if they drive in and out once a day to get to and form work, or try to combine trips to minimize use, that would be "normal wear and tear," but if they hire a contractor to put an addition on their home during mud season instead of waiting a few weeks, and there are multiple trucks coming and going daily, that would not be "normal wear and tear." 

Some other options would be to remind residents before the frost goes out to get their oil tanks/propane tanks etc filled while the road will support the trucks, and to be judicious about their use of the road during mud season.  Often, it's possible to make a trip out for groceries early in the morning while the road is still frozen and get back before it thaws.

Of course the ideal solution would be to do whatever maintenance is needed to bring the road to a condition where mud season is not a problem.  Make sure that culverts are clean and in good repair, that ditches provide adequate drainage, and that there is sufficient gravel to float vehicles over the mud.  Even if you can't do that all at once, try to do a bit more each year until the road is up to standard.  I've probably missed something, so be creative - think outside the box.  I'd love to hear how others have addressed such problems.