Yes it's clear you can use willow, but as mentioned they won't last long, so it depends on whether you want practicality or aesthetics. Take a look at what Charles Dowding does, he advocates putting cardboard down onto grass with compost on top. He puts a wooden frame round but reccomends removing them after a couple of years and just having a low mounded beds.
I like your idea. Noticed Amazon are selling a pack of 5 (120cm x 20cm) willow hurdles for lawn edging for under £60, think that's comparable with wood, but maybe not lasting as long?
Fencing timber is only a few quid per metre, at most. Completely different from those pre-cut timber beds which are always very expensive. All you need is a few battens or some 3x2, or similar, for the corners to fasten it to. If it's a higher bed you can use fence posts and concrete them in, but I'm guessing that wouldn't be needed as it's just for veg.
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Take a look at what Charles Dowding does, he advocates putting cardboard down onto grass with compost on top. He puts a wooden frame round but reccomends removing them after a couple of years and just having a low mounded beds.
Cambridgeshire/Norfolk border.
If it's a higher bed you can use fence posts and concrete them in, but I'm guessing that wouldn't be needed as it's just for veg.
This is the sort of thing I use [a fairly local outlet to me]
https://www.jwgrant.co.uk/wallplate-sawn-unseasoned-fencing-timber-treated-100-x-22mmosaw07
You can cut it very easily to length, and even if you need it delivered, it's usually still cheaper. Any builders' merchant will have similar products.
I live in west central Scotland - not where that photo is...