My project this year was going to be a plastic free allotment. It's not quite working out how I planned, for various reasons, but here's some of my methods
Soil blocks and paper pots to start plants off. I like bamboo pots, and silicone module trays, but they are expensive.
Wooden beds with metal archways and chicken wire instead of nets
cardboard and woodchip mulch to suppress weeds (I'm trying to do without landscape fabric where I can)
Biological control for pests where possible (slugs and caterpillars, rather than plastic barriers and nets)
chicken wire or jute nets for climbers
I'm using plastic compost bins, water butts, buckets and cans at the moment, because I have them, but I will replace with metal or wood as they wear out.
I can't think of anything but fleece for carrot root fly though.