I'm afraid the Amalfi Coast
has long been extremely popular, initially with Italians themselves and now with (it often seems) the whole world....and pretty-much year-round. If you hope to avoid crowds (though not other visitors) the winter months are the only feasible option.
For exactly that reason I've visited three times in February and once in late October, basing myself in Sorrento
. Imo Sorrento makes the best base out of season because it's bigger than Amalfi
and, whilst a still a very visitor-popular town, it's not quite so tourist-focused. It's an ordinary working town so the vast majority of its shops, restaurants, hotels etc remain open year-round. Accessing the Amalfi Coast settlements from Sorrento is easy using the regular, pretty frequent bus service.
I haven't encountered any 'sights' (e.g. archaeology, history, museums and churches) being closed though opening hours of course change during the winter months. Ferries from Sorrento to Capri (wonderful walking out of season) and Naples run all year round.
The weather on all my visits has been perfectly ok. No hot sunshine, of course, but plenty of warm sunshine and clear days. I had only one day of chilly rain during all my visits. Imo the biggest downside of visiting in the late Autumn & winter months is the shorter hours of daylight rather than the weather.
So I suggest you look at the last week of October (though it's still shoulder season) through to early April, depending on the Easter date. Avoid the weeks leading up to and after Christmas and New Year and the weeks before and after Easter: all are a sort of mini-high-season, with visitor numbers and hotel rates to match.