Fixmyspeaker.com isn't a buildhouse.
They simply recones sub.
Also, a few years ago, RD Audio claimed to wind there own coils. But it turned out to be a lie.
But youtube the how its made, loudspeaker. There is a short clip in that video of a voicecoil being made.
But IMO, its not as easy as pie. And going to a coil buildhouse would be simpler, and cost less.
Precision Ecowound(spelling) is one of the best coil makers I believe.
There's another one, a sister company that Precision but I can't think of its name.