Yes, the winemakers in Alsace are open all year. (I used to live in a winemaking village there.) Most are closed on Sundays. Do not even think about turning up on Good Friday.
At most places, except perhaps the very top names (Trimbach, Dopff & Irion, Zind-Humbrecht), you can often turn up on spec and just ask for a tasting. This is normally free. The contrepartie is that you are expected to be sincerely interested in buying 6 bottles, preferably 12. If you arrive in a car this will be just assumed. The more serious you appear, the better the welcome. "Hi, we're looking for a couple of dozen bottles for a wedding" will go down great, as will "We'll definitely have six of that one" after one of the first few glasses. To be fair there aren't many timewasters driving around, but most places have their regulars, and there are not too many English-speaking tourists turning up at random.
Be aware that a full Alsace tasting may involve a dozen or more different wines. There are seven main grapes, plus most places have an Edelzwicker (blend), and then there's the crémant which may have a non-standard grape like Chardonnay in it. Next you get to the grands crus (€€€), the vendanges tardives (€€€€) and the sélections de grains nobles (€€€€€).
Even though the tasting servings are very small, a full session of the whole range will easily put you over the 0.05 drink-drive limit. Plan accordingly (e.g., taste at 4:30pm having checked in to your hotel a kilometre walk away). One time I went with a friend to the least highly-rated producer in our village. DP wisely stayed at home. The tasting was run by Grandma, who was babysitting a 5-year-old. We learned some interesting Alsatian dialect vocabulary, and we also got completely plastered as she was pouring at least 10cl glasses!
Don't hesitate to go into a cooperative. Unlike much of France, in Alsace the co-op wines can be every bit as good as those from places that make wine from their own grapes.