Is it possible to have healthy plants (edible ones) flourishing in pots without the use of synthetic fertilisers? I have concerns regarding fertilisers polluting nearby water sources/soil from drainage as well as sustainability.
If you want to avoid both synthetic fertilizers and animal waste (as discussed in some of the comments), there are some organic fertilizers that are based on waste products that would have to be disposed of otherwise.
The Bokashi method for example turns kitchen waste into liquid fertilizer on a household scale.
Some commercial producers also claim that their product uses waste from food production, e.g. grains from brewing. They are probably the most convenient as they are standardized in their strength and NPK ratio and tested for that during production.
Some gardeners apply a bit of their own compost or fermented plant “teas”, but that can be a bit iffy for indoor plants if not done very well - for example it can be a bit smelly.
I recommend you do a bit of research on what exactly your potted plants need and then fertilize accordingly.
Yes that is possible, you can use organic fertilizer instead. For example cow or chicken manure can be used. I have these in dried granules form, I just add a hand full every month or so when I think the plants need some extra nutrients. But there are more organic fertilizers, check here on wiki if you want.
There are a lot of organic fertilizers both solid and liquid you can use and they are more than enough to produce great crops.
The amount and type of fertilizer depends on the plant though so you are going to have to do some research on your own.
The main nutrients are NPK - Nitrogen, Phosphorus and Potassium - and you can run into trouble with lack of some trace minerals but that shouldn't be that big of a problem.
Most fertilizers have the ratios of NPK on their label somewhere. Very simplified Nitrogen is for foliage, Phosphorus for flowers and fruit and Phosphorus makes the cells strong and is important for photosynthesis and the building of starches and sugars.
For solid fertilizers you can get or make fertilizers based manure based, wool, bone, blood, algae, pomace and lots of other materials. Liquid you can use vinasse or plant teas or slurrys - most popular based on stinging nettle - or compost based teas.
For most plants regular compost should work well enough too.
Also a lot of potting mix comes pre fertilized. That should be enough for most vegetables in the beginning.
Easiest way is to look up if, what kind and how much fertilizer you need for what you want to grow. There should be something organic that's close enough. Don't get too hung up on details though, plants are in general pretty forgiving and gardening means always learning.
Barrel composting of vegetable leftovers provides me several buckets of composted material each spring. I believe the ~20 liter buckets I have on my balcony don't get hot enough to kill pathogens ... but haven't had problems yet. I imagine a 120L barrel seeing some sun could reach suitable temperatures ... but then you can't have worms anymore.