Do I have to let my town use my private access road while they close another road for repairs?

You should discuss this with your town. Some of your neighbors may have the right to use it, if their only other access to the broader road network is cut off by the road shutdown. However, for general traffic, you may have the right to recompense for the use of the land. If you get significant pushback, then you should call an attorney to discuss the issue, as they can help you negotiate and threaten appropriate legal action if they attempt to force the issue. But you never know, they could also agree to a reasonable payment for use right off the bat. Stranger things have happened.