We just went to the Union ice rink in Woodstock. People use Walkers there with wheels to learn to skate so if those don’t do serious damage to the ice then the same would apply to a stroller. If there is anyone around from the town ask them otherwise I would just go ahead and use it after wiping the Wheels off first. If anyone objects (which I doubt they will unless you are zooming around knocking into people at high speed) then you can plead your case then or leave.