Is it possible to have the wood stuck together for such a bowl without using any adhesive
If you do it right, yes. This is the foundation of much traditional bucket and barrel making all over the world, the idea being that by shaping the staves with the required angled edges and then squeezing them together from the outside with binding of some sort1 the staves are forced to remain together.
However, while it's theoretically possible to make a hangiri without any adhesive some glue is used in their construction.
There are actually numerous videos on YouTube showing similar storage containers being made by Japanese crafstmen. For hangiri specifically the best of the ones I've watched is this on the Woodworking Enthusiasts channel. In addition to what many similar videos show, various aspects of the build and finish-shaping processes, it also gives some key details of wood selection which others seem to skip over (unless this is in the Japanese commentary). Watch your knuckles planing the edges of the staves like this! :-)
Since all the toxicity related stuff is really secondary to what you're asking about I'll cover it separately.
Most of the adhesives have VOCs in them
Everyone's level of worry about toxicity is their own, but it should be based on a realistic assessment of the risk. This should be especially the case for those living in the US, where sheet goods are used so heavily in both commercial and residential construction.
So, for a product like this, first remember the ancient adage, "the dose determines the poison". This isn't just a snappy saying2.
And here, if you did use a modern glue, how much is actually present when you're done? Well a good invisibly thin glue line is a thousandth of an inch at most..... I'll leave you to do the math on what the contact area actually is2 o_O
I find it weird and disgusting to keep wet rice to cool down in an adhesive-based wood composite.
Do you similarly find it weird or disgusting to cut food on an "adhesive-based wood composite" or are all your cutting boards single boards? Or solid plastic or resin? It's worth pointing out that all boards made from solid plastic or resin, or faced with melamine (which seems to cover 99% of boards that aren't wood from what I've seen) each contain their own set of VOCs.
any wood-based glue (e.g. with wood aroma or sap aromas)?
FYI such an adhesive would contain its own VOCs! They may be natural, but that doesn't at all make them harmless. Perhaps the most famous wood-derived extract is turpentine, which despite being natural is more toxic than most mineral spirits.
1 It doesn't have to be copper strips as in the originals if you find it difficult to source or too costly, you could use wooden strips in a similar manner to the way that Shaker boxes are constructed. Stainless steel wire seems like it would also be a viable alternative without any worries about corrosion.
2 A good example of the principle is that apple seeds actually contain a deadly toxin — you think that's almond you're tasting? ;-) — but despite this it's still perfectly safe to eat the core of an apple.
3 A useful, and quite direct, comparison is we (most people) don't generally worry about food, hot or cold, in contact with the glue joints in the vast majority of wooden cutting boards, whether commercially produced or homemade, made for the past 50+ years.....