Unfortunately, you cannot simply go without clothes in a multigenerational ship. Even apart from considerations of supportive comfort (bras, underpants) consider that even very basic tasks like cooking and cleaning can't really be done naked without the danger of burns (chemical or heat based) to sensitive parts of your anatomy. Now consider that on board a generation ship you may be required to tend to a nuclear reactor, repair a maintenance robot with a welder, sort out that pesky oil leak on the shuttle, wade through a flooded sewage system or even go and clean the exterior windows on the bridge, and you can see the issue!
Clothing wise, why not recycle as much as possible? On a generation ship you either have sufficiently advanced handwavium based technology to create matter out of energy (a la Star Trek's replicators) or you have vast swaths of spaceship devoted to farming, food processing and other industries vital to surviving. Therefore, you're looking at either anything you damn well please in the first case, or animal and plant based in the latter.
In the first case, a repli-wardrobe would create your clothes from a set of templates, with minor alterations dependant on tailoring, fashion and (in the case where energy might be limited for personal use) cost. they would be worn as normal, and either recycled back into energy at the end of the day, or laundered as you would on earth.
In the latter case, a small proportion of the generation ship's compliment would be devoted to creating clothes for the ship's crew. Laundering need not be a group effort unless your generation ship's society is significantly collectivised. We all manage to launder our own clothes here on earth, after all :)
In both cases, they can all be recycled once they've reached their useful life either as energy or into things like painters rags or insulation. Waste-not-want-not is the watchword in a closed system.
Just another quick thought. Synthetics like Polyester are oil based, which precludes them. You might be able to run up something using corn starch or something similar, but I'm not a chemical engineer so I don't know anything more about that!
Therefore, your options in a non-replicator based system are essentially plant materials or animal materials, dependant on what's being grown aboard ship.
We can make synthetic oil from elements, given an energy input
(Fischer-Tropsch). Or there's Rayon, which is made from plant
cellulose. Some "biodegradable" plastic bags are made from corn, but
they are intended to disintegrate rapidly - pjc50
There you go: plant material, animal material or synthetic oil from elements with the right input would mean it really depends on the type of generation ship you're creating :)