I have question similar to
what to look for when buying bicycle pants
Are Merino cycling boxers/cycling pants worth the cost?
Finding good pants for winter biking
...but they all speak about padding and shape and do not discuss the fabric. I have been wearing only cotton (also linen a couple times) throughout my life and am trying to avoid materials which are not biodegradable. In the case polyester is not the only option, I would like to know what natural materials you could recommend.
I assume here that you want clothing designed for cycling, or at least for sports, as much as possible, so for significant commuting, leisure, or fast rides, but not just wandering around town, as that could be done in everyday clothes. I deliberately haven't linked to products, as I don't want to imply any endorsement. I've also mentioned clothing for other areas, as I came across things while I was looking into your main question and they're useful for the broader underlying question.
Merino wool is well known in cycling and other outdoor activities, especially as a base layer and for socks. You can buy jerseys quite easily but there's not much for your lower half
Bamboo fibre is almost always rayon, almost pure plant cellulose. Bike shorts and many other forms of fitness clothing are available. As cellulose, it should be biodegradable (over long periods I suspect) and it's from potentially sustainable sources. Other forms of rayon may also be suitable. I've had socks made from it in the past; though not specifically for cycling they seemed reasonable even when my feet got wet.
Gloves may be tricky. Leather is your best bet to keep the wind off and provide some protection from water, perhaps oversized to go over wool in the winter. Suede is good for gripping. Any padding won't meet your requirements - so comfort may be an issue.
A waterproof top layer might be tricky, unless you go for a waxed cotton jacket, but they tend to be bulky and heavy, and not designed for cycling.
For shoes, you'll also struggle. Leather bike shoes exist but are uncommon and by the time leather has been tanned using modern methods it's almost a petrochemical product held together with a bit of cow. Soles etc. are universally synthetic. If you really want biodegradable shoes you can probably get some espadrilles with cotton uppers - they'd even look quite stylish on a vintage Pashley, but you're not going to get cleats on them.
However if you're worried about microplastic pollution, shoes are much less of an issue as you don't wash them very often, and washing is when they shed. This also applies to waterproofs. When you've completely worn something out, you can dispose of it properly (yes, landfill, but a lot of textiles end up there whatever they're made of).
If biodegradable is a requirement than the conversation is going to end at wool, at least for riding applications where staying warm when it's wet/cold is a factor. The big problem with non-cycling-specific wool lowers in my experience is the propensity to blow out seams or wear through material in the crotch, both of which are only patchable for so long without more radically reworking the garment (which may be a reasonable thing to do depending on your priorities.) Wool cycling pants, tights, and shorts are great if the trim/tight/sporty cuts they mostly all come in work for you.
Likewise for shorts/tights to wear under your wool pants or solo, wool pieces with traditional leather chamoises are still available and are kind of quietly the best thing going anyway if you can accept the cost.
