It seems like one way to do this is to use several spells:
Level 15: Base 5 MuVi (maintain a concentration spell of up to level 15); D: Ring +2, Target: Circle +0
Level 10: Base 5 MuAn (minor unnatural change: Produces only non-sticky webs, of a specific color), D: Concentration +1, Target: Circle +0
Level 3: Base 2 ReAn (plant a single suggestion in the mind of an animal: Only houseflies are worth killing / eating), D: Concentration +1, Target: Circle +0
Level 3: Base 2 ReAn (plant a single suggestion in the mind of an animal: Wrapping my thread around this spool is much better than building an ordinary web), D: Concentration +1, Target: Circle +0
Level 3: Base 2 ReAn (plant a single suggestion in the mind of an animal: Never leave the Ring), D: Concentration +1, Target: Circle +0
Level 10: Base 5 MuAn (minor unnatural change: Spider is marked as a 'production spider' by having its colors & pattern changed to vivid colors of the covenant), D: Concentration +1, Target: Circle +0
Level 10: Base 5 ReAn (paralyze an animal: any live vermin (which are not 'production spiders') which touches the inside of the production tank is paralyzed as long as it does so), D: Concentration +1, Target: Circle +0
Level 10: Base 5 PeAn (Destroy an animals corpse: any dead vermin (flies, spiders, mice, etc) which touch the inside of this jar are destroyed), D: Concentration +1, Target: Circle +0
Level 10: Base 5 ReAn (paralyze an animal); D: Diameter +1, T: Circle +0 -- paralyze (for two minutes / 20 rounds) any spiders in a circle. Enough time to get it into a jar for transport, or to place it in the 'farm' while you do the spells to get it into production.
These spiders are under a constant minor magical effect, so accumulate 1 warping per year -- which is longer than spiders live in any event. They need to be fed, the spools changed, and you will need a different tank for each color thread -- but the colored thread is non-magical; the animal which produced it was magically altered to make this the natural color.
There are probably better ways to do this; as each new batch of production spiders (all of them within the 'Circle' target) will take several castings, each followed by the MuVi spell at the top. Giant spiders which are shrunk down are an easier-to-manage size, and their thread is full sized --- but they eat more, too, and the resize spells might be hard. I guess you would need to feed them giant houseflies which had been shrunk down as well.
None of these spells are particularly advanced; this is something that could be relegated to whichever apprentices who could manage the MuVi spell.